16. Q16c
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Similar Question 1
<p>Find each product or quotient.</p><p><code class='latex inline'>\displaystyle \frac{\sqrt{x^{4} y}}{\sqrt[4]{x^{2} y^{8}}} </code></p>
Similar Question 2
<p>Compare the two numbers. Use <code class='latex inline'>\displaystyle > </code> or <code class='latex inline'>\displaystyle < </code> </p><p><code class='latex inline'>\displaystyle \sqrt{63}, 7.5 </code></p>
Similar Question 3
<p>Simplify each expression.</p><p><code class='latex inline'>\displaystyle \frac{\sqrt[8]{81}}{\sqrt[6]{3}} </code></p>
Similar Questions
Learning Path
L1 Quick Intro to Factoring Trinomial with Leading a
L2 Introduction to Factoring ax^2+bx+c
L3 Factoring ax^2+bx+c, ex1
Now You Try
<p>Simplify each expression.</p><p><code class='latex inline'>\displaystyle \sqrt{(-2)^{2}+8^{2}} </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle \sqrt{48} </code></p>
<p>In the expression <code class='latex inline'>\displaystyle \sqrt[n]{x^{m}}, m </code> and <code class='latex inline'>\displaystyle n </code> are positive integers and <code class='latex inline'>\displaystyle x </code> is a real number. The expression can be simplified.</p><p>a. If <code class='latex inline'>\displaystyle x > 0 </code>, what are the possible values for <code class='latex inline'>\displaystyle m </code> and <code class='latex inline'>\displaystyle n ? </code></p><p>b. If <code class='latex inline'>\displaystyle x < 0 </code>, what are the possible values for <code class='latex inline'>\displaystyle m </code> and <code class='latex inline'>\displaystyle n ? </code></p><p>c. If <code class='latex inline'>\displaystyle x < 0 </code>, and an absolute value symbol is needed in the simplified expression, what are the possible values of <code class='latex inline'>\displaystyle m </code> and <code class='latex inline'>\displaystyle n ? </code></p>
<p>Find all the real square roots of each number.</p><p>225</p>
<p>Compare <code class='latex inline'>\displaystyle \sqrt{26} </code> and 4.9. Explain your answer.</p>
<p>Simplify. <code class='latex inline'>\displaystyle \sqrt{200 a^{6} b^{7}} </code></p>
<p>Write each expression in radical form, or write each radical in exponential form.</p><p><code class='latex inline'>\displaystyle \sqrt[4]{7 x^{6} y^{9}} </code></p>
<p>Rationalize each denominator.</p><p><code class='latex inline'>\displaystyle \frac{5\sqrt{3}}{2\sqrt{3} + 4} </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle \sqrt{68} </code></p>
<p>Simplify each radical expression. </p><p><code class='latex inline'>\displaystyle \sqrt{\frac{15 x}{x^{3}}} </code></p>
<p>Find all the real fourth roots of each number.</p><p><code class='latex inline'>16</code></p>
<p>Simplify. <code class='latex inline'>\displaystyle \sqrt[3]{54 y^{10}} </code></p>
<p>Write each expression in exponential form.</p><p><code class='latex inline'>\displaystyle \sqrt[4]{c^{2}} </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle (4- \sqrt{3})(5+2\sqrt{3}) </code></p>
<p>Find each real root.</p><p><code class='latex inline'>\displaystyle \sqrt{0.25} </code></p>
<p>Rationalize each numerator.</p><p><code class='latex inline'>\displaystyle \frac{2\sqrt{3} + \sqrt{7}}{5} </code></p>
<p>Evaluate. </p><p><code class='latex inline'>\displaystyle \sqrt{169} </code></p>
<p>Evaluate. </p><p><code class='latex inline'>\displaystyle \sqrt{0.49} </code></p>
<p>Compare the two numbers. Use <code class='latex inline'>\displaystyle > </code> or <code class='latex inline'>\displaystyle < </code> </p><p><code class='latex inline'>\displaystyle 4, \sqrt{12} </code></p>
<p>Simplify each radical expression. Use absolute value symbols when needed.</p><p><code class='latex inline'>\displaystyle \sqrt[4]{\frac{1}{256}} </code></p>
<p>Error Analysis A student simplified the radical expression at the right. What mistake did the student make? What is the correct answer?</p><p><code class='latex inline'>\displaystyle \sqrt{\frac{5 x}{25}}-\frac{\sqrt{5}}{5} x </code></p>
<p>Simplify each expression.</p><p><code class='latex inline'>\displaystyle \frac{\sqrt[5]{64}}{\sqrt[5]{4}} </code></p>
<p>Rationalize each denominator. Simplify your answer.</p><p><code class='latex inline'>\displaystyle \frac{4}{3 \sqrt{3}-2} </code></p>
<p>Add or subtract.</p><p><code class='latex inline'>\displaystyle \frac{4}{\sqrt{5}-\sqrt{3}}-\frac{4}{\sqrt{5}+\sqrt{3}} </code></p>
<p>Rationalize each numerator.</p><p><code class='latex inline'>\displaystyle \frac{\sqrt{3} - \sqrt{7}}{4} </code></p>
<p>Simplify each radical expression if <code class='latex inline'>\displaystyle n </code> is even, and then if <code class='latex inline'>\displaystyle n </code> is odd.</p><p><code class='latex inline'>\displaystyle \sqrt[n]{m^{4n}} </code></p>
<p>Simplify each radical expression. Use absolute value symbols when needed.</p><p><code class='latex inline'>\displaystyle \sqrt{16 x^{2}} </code></p>
<p>Explain the error in this simplification of radical expressions.</p><img src="/qimages/22962" /><p><code class='latex inline'>\displaystyle \begin{aligned} \frac{\sqrt[7]{x^{5}}}{\sqrt[4]{x^{2}}} &=7-4 \sqrt{\frac{x^{5}}{x^{2}}} \\\\ &=\sqrt[3]{x^{5-2}} \\\\ &=\sqrt[3]{x^{3}} \\\\ &=x \end{aligned} </code></p>
<p>Multiply.</p><p><code class='latex inline'>\displaystyle (5+2 \sqrt{5})(7+4 \sqrt{5}) </code></p>
<p>Find each real root.</p><p><code class='latex inline'>\displaystyle \sqrt[5]{-243} </code></p>
<p>Determine whether each expression is always, sometimes, or never a real number. Assume that <code class='latex inline'>\displaystyle x </code> can be any real number.</p><p><code class='latex inline'>\displaystyle \sqrt[3]{-x^{2}} </code></p>
<p>Is each equation always, sometimes, or never true? Explain your answer.</p><p><code class='latex inline'>\displaystyle \sqrt[3]{x^{8}}=x^{2} </code></p>
<p>Is each equation always, sometimes, or never true? Explain your answer.</p><p><code class='latex inline'>\displaystyle \sqrt{x^{6}}=x^{3} </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle 2 \sqrt{6}(3 \sqrt{6}-5 \sqrt{8}) </code></p>
<p>Simplify each radical expression if <code class='latex inline'>\displaystyle n </code> is even, and then if <code class='latex inline'>\displaystyle n </code> is odd.</p><p><code class='latex inline'>\displaystyle \sqrt[n]{m^{n}} </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle \sqrt{20 x^{3}} </code></p>
<p>Simplify each expression.</p><p><code class='latex inline'>\displaystyle \frac{8^{\frac{1}{6}}-9^{\frac{1}{4}}}{\sqrt{3}+\sqrt{2}} </code></p>
<p>You can define the rules for irrational exponents so that they have the same properties as rational exponents. Use those properties to simplify each expression.</p><p><code class='latex inline'>\displaystyle \frac{x^{4 \pi}}{x^{2 \pi}} </code></p>
<img src="/qimages/60542" /><p>Error Analysis A student simplified an expression, as shown below. Describe and correct the error. <code class='latex inline'>\displaystyle \frac{1}{\sqrt{3}-1}=\frac{1}{\sqrt{3}-1} \cdot \frac{\sqrt{3}+1}{\sqrt{3}+1}-\frac{\sqrt{3}+1}{9-1}=\frac{\sqrt{3+1}}{8} </code></p>
<p>Simplify if possible.</p><p><code class='latex inline'>\displaystyle 5 \sqrt{6}+\sqrt{6} </code></p>
<p>Find each product or quotient.</p><p><code class='latex inline'>\displaystyle (\sqrt[4]{6})(\sqrt[3]{6}) </code></p>
<p>Determine whether each expression is always, sometimes, or never a real number. Assume that <code class='latex inline'>\displaystyle x </code> can be any real number.</p><p><code class='latex inline'>\displaystyle \sqrt{-x^{2}} </code></p>
<p>Estimate the square root. Round to the nearest integer.</p><p><code class='latex inline'>\displaystyle \sqrt{35} </code></p>
<p>Calculate the product of each radical expression and its corresponding conjugate.</p><p>a. <code class='latex inline'>\displaystyle \sqrt{5} - \sqrt{2} </code></p><p>b. <code class='latex inline'>\displaystyle 3\sqrt{5} + 2\sqrt{2} </code></p><p>c. <code class='latex inline'>\displaystyle 9 + 2\sqrt{5} </code></p><p>d. <code class='latex inline'>\displaystyle 3\sqrt{5} -2\sqrt{10} </code></p>
<p>You can define the rules for irrational exponents so that they have the same properties as rational exponents. Use those properties to simplify each expression.</p><p><code class='latex inline'>\displaystyle \left(7^{\sqrt{2}}\right)^{\sqrt{2}} </code></p>
<p>Express each radical as a power.</p><p>a) <code class='latex inline'>\displaystyle \sqrt{x} </code></p><p>b) <code class='latex inline'>\displaystyle \sqrt[3]{x} </code></p><p>c) <code class='latex inline'>\displaystyle (\sqrt[4]{x})^3 </code></p><p>d) <code class='latex inline'>\displaystyle \sqrt[5]{x^2} </code></p>
<p>Find all the real cube roots of each number.</p><p><code class='latex inline'>\displaystyle -64 </code></p>
<p>Write each expression in exponential form.</p><p><code class='latex inline'>\displaystyle (\sqrt{7 x})^{3} </code></p>
<p>Convert <code class='latex inline'>\displaystyle 49^{2.5} </code> into radical form and evaluate the expression.</p>
<p>Rationalize each denominator.</p><p><code class='latex inline'>\displaystyle \frac{2\sqrt{3}}{\sqrt{3} -2} </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle (1+\sqrt{72})(5+\sqrt{2}) </code></p>
<p>Simplify. <code class='latex inline'>\displaystyle \sqrt{50 x^{5}} </code></p>
<p>Evaluate without a calculator.</p><p><code class='latex inline'>\displaystyle \sqrt{1.96} </code></p>
<p>Evaluate, to the nearest tenth. </p><p><code class='latex inline'>\displaystyle \sqrt{1253} </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle \sqrt[4]{32}+\sqrt[4]{48} </code></p>
<p>Find each real root.</p><p><code class='latex inline'>\displaystyle \sqrt[3]{-27} </code></p>
<p>Estimate the square root. Round to the nearest integer.</p><p><code class='latex inline'>\displaystyle \sqrt{320} </code></p>
<p>Write each expression in radical form, or write each radical in exponential form.</p><p><code class='latex inline'>\displaystyle \sqrt[4]{625 x^{2}} </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle \sqrt[3]{54}+\sqrt[3]{16} </code></p>
<p>Find all the real fourth roots of each number.</p><p><code class='latex inline'>0.0081</code></p>
<p>Simplify each expression. Rationalize all denominators.</p><p><code class='latex inline'>\displaystyle \sqrt{3 x} \cdot \sqrt{5 x} </code></p>
<p>Multiply each pair of conjugates.</p><p><code class='latex inline'>\displaystyle (5-\sqrt{11})(5+\sqrt{11}) </code></p>
<p>Express as a mixed radical in simplest form.</p><p> <code class='latex inline'>\displaystyle \sqrt{147} </code></p>
<p>Find each real root.</p><p><code class='latex inline'>\displaystyle -\sqrt[3]{64} </code></p>
<p>Evaluate. Expression in rational form(without negative exponents).</p><p><code class='latex inline'>\displaystyle \sqrt[5]{-32}(\sqrt[6]{64})^5 </code></p>
<p>Simplify.</p><p><code class='latex inline'> \displaystyle 9\sqrt{7} - 4\sqrt{7} </code></p>
<p>Rationalize each numerator.</p><p><code class='latex inline'>\displaystyle \frac{2\sqrt{3} -5}{3\sqrt{2}} </code></p>
<p>Compare the two numbers. Use <code class='latex inline'>\displaystyle > </code> or <code class='latex inline'>\displaystyle < </code> </p><p><code class='latex inline'>\displaystyle -\sqrt{3},-\sqrt{5} </code></p>
<p> Simplify the result to express the surface area of a cylinder with a height of 10 m in terms of its volume.</p>
<p>Simplify each expression.</p><p><code class='latex inline'>\displaystyle \sqrt[8]{36 h^{4} j^{4}} </code></p>
<p>Write each expression in radical form, or write each radical in exponential form.</p><p><code class='latex inline'>\displaystyle \sqrt{17} </code></p>
<p>Simplify each expression.</p><p><code class='latex inline'>\displaystyle \frac{\sqrt[8]{81}}{\sqrt[6]{3}} </code></p>
<p>a. Suppose <code class='latex inline'>\displaystyle n </code> is an even number. Simplify <code class='latex inline'>\displaystyle \sqrt{x^{n}} </code>.</p><p>b. Suppose <code class='latex inline'>\displaystyle n </code> is an odd number greater than <code class='latex inline'>\displaystyle 1 . </code> Simplify <code class='latex inline'>\displaystyle \sqrt{x^{n}} </code>.</p>
<p>Simplify each expression. Rationalize all denominators.</p><p><code class='latex inline'>\displaystyle \sqrt[3]{2 x^{2}} \cdot \sqrt[3]{4 x} </code></p>
<p>Multiply, if possible. Then simplify.</p><p><code class='latex inline'>\displaystyle \sqrt[3]{9}\sqrt[3]{-81} </code></p>
<p>Write each expression in exponential form.</p><p><code class='latex inline'>\displaystyle (\sqrt[3]{a})^{2} </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle 7 \sqrt{24}+3 \sqrt{28}+9 \sqrt{54}+6 \sqrt{175} </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle \sqrt{72}+\sqrt{32}+\sqrt{18} </code></p>
<p>Simplify. <code class='latex inline'>\displaystyle \sqrt[3]{81 x^{3}} </code></p>
<p>Divide and simplify.</p><p><code class='latex inline'>\displaystyle \frac{\sqrt[3]{48 x^{3} y^{2}}}{\sqrt[3]{6 x^{4} y}} </code></p>
<p>Simplify each radical expression. </p><p><code class='latex inline'>\displaystyle 3 \sqrt{5 m} \cdot 4 \sqrt{\frac{1}{5} m^{3}} </code></p>
<p>For a linear equation in standard form <code class='latex inline'>\displaystyle A x+B y=C </code>, where <code class='latex inline'>\displaystyle A \neq 0 </code> and <code class='latex inline'>\displaystyle B \neq 0 </code>, the distance <code class='latex inline'>\displaystyle d </code> between the <code class='latex inline'>\displaystyle x </code> - and <code class='latex inline'>\displaystyle y </code> -intercepts is given by <code class='latex inline'>\displaystyle d=\sqrt{\left(\frac{C}{A}\right)^{2}+\left(\frac{C}{B}\right)^{2}} \cdot </code> What is the distance between the <code class='latex inline'>\displaystyle x </code> - and <code class='latex inline'>\displaystyle y </code> -intercepts of the graph of <code class='latex inline'>\displaystyle 4 x-3 y=2 ? </code></p>
<p>Simplify each expression.</p><p><code class='latex inline'>\displaystyle \sqrt[4]{9 g^{2}} </code></p>
<p>Simplify each radical expression. Use absolute value symbols when needed.</p><p><code class='latex inline'>\displaystyle \sqrt[3]{0.125} </code></p>
<p>Simplify each expression.</p><p><code class='latex inline'>\displaystyle \sqrt[6]{216} </code></p>
<p>Simplify each radical expression.</p><p><code class='latex inline'>\displaystyle \sqrt{a^{8} b^{18}} </code></p>
<p>Determine whether the following statement is true or false. Provide a proof or counterexample to support your answer. <code class='latex inline'>\displaystyle x+y > \sqrt{x^{2}+y^{2}} \text { when } x > 0 \text { and } y > 0 </code></p>
<p>Multiply.</p><p><code class='latex inline'>\displaystyle (\sqrt{3}+\sqrt{5})^{2} </code></p>
<p>Convert <code class='latex inline'>\displaystyle (-27)^{\frac{5}{3}} </code> into radical form and evaluate the expression.</p>
<p>Simplify each product.</p><p><code class='latex inline'>\displaystyle 10 \sqrt{12 x^{3}} \cdot 2 \sqrt{6 x^{3}} </code></p>
<p>Name the subset(s) of the real numbers to which each number belongs.</p><p><code class='latex inline'>\sqrt{11}</code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle \frac{1}{2} \sqrt{180}-\frac{6}{7} \sqrt{245}+\frac{2}{3} \sqrt{405} </code></p>
<p>Rationalize the denominators and simplify.</p><p><code class='latex inline'>\displaystyle \frac{4-2 \sqrt[3]{6}}{\sqrt[3]{4}} </code></p>
<p>Compare the two numbers. Use <code class='latex inline'>\displaystyle > </code> or <code class='latex inline'>\displaystyle < </code> </p><p><code class='latex inline'>\displaystyle -4,-\sqrt{4} </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle \sqrt{180} </code></p>
<p>Does <code class='latex inline'>\displaystyle \sqrt{4^{3}}-\sqrt{4}=4 ? </code> Explain why or why not.</p>
<p>Multiply each pair of conjugates.</p><p><code class='latex inline'>\displaystyle (4-2 \sqrt{3})(4+2 \sqrt{3}) </code></p>
<p>Find each product or quotient.</p><p><code class='latex inline'>\displaystyle \frac{\sqrt[6]{4}}{\sqrt[3]{4}} </code></p>
<p>Simplify each expression.</p><p><code class='latex inline'>\displaystyle \sqrt{23} \cdot \sqrt[3]{23^{2}} </code></p>
<p>Simplify each expression. Rationalize all denominators.</p><p><code class='latex inline'>\displaystyle \frac{\sqrt{32}}{\sqrt{2}} </code></p>
<p>Simplify each radical expression. Use absolute value symbols when needed.</p><p><code class='latex inline'>\displaystyle \sqrt[4]{16 c^{4}} </code></p>
<p>The formula <code class='latex inline'>\displaystyle r=\sqrt{\frac{d}{p}}-1 </code> gives the interest rate <code class='latex inline'>\displaystyle r </code>, expressed as a decimal, that will allow principal <code class='latex inline'>\displaystyle P </code> to grow into amount <code class='latex inline'>\displaystyle A </code> in <code class='latex inline'>\displaystyle 2 \mathrm{yr} </code>, if the interest is compounded annually. If you invest <code class='latex inline'>\displaystyle \$ 10,000 </code> and want to make <code class='latex inline'>\displaystyle \$ 2000 </code> in interest over 2 yr, what interest rate do you need?</p> <ul> <li><p>What amount do you want in the account after 2 yr?</p></li> <li><p>What radical expression gives the interest rate you need?</p></li> </ul>
<p>Find all the real square roots of each number.</p><p><code class='latex inline'>\displaystyle \frac{64}{169} </code></p>
<p>Simplify each expression. Rationalize all denominators.</p><p><code class='latex inline'>\displaystyle 5 \sqrt{2 x y^{6}} \cdot 2 \sqrt{2 x^{3} y} </code></p>
<p>Reasoning Describe the possible values of <code class='latex inline'>\displaystyle a </code> such that <code class='latex inline'>\displaystyle \sqrt{72}+\sqrt{a} </code> simplifies to a single term.</p>
<p>Order the numbers from least to greatest.</p><p><code class='latex inline'>\displaystyle -1.5,-0.5,-\sqrt{2},-1.4 </code></p>
<p>Write each expression in radical form, or write each radical in exponential form.</p><p><code class='latex inline'>\displaystyle \sqrt[3]{15} </code></p>
<p>Rationalize each denominator.</p><p><code class='latex inline'>\displaystyle \frac{3\sqrt{2}}{2\sqrt{3} - 5} </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle -3\sqrt{75} </code></p>
<p>Use a calculator to solve the equation <code class='latex inline'>\displaystyle 7^{2 x}=75 . </code> Round the answer to the nearest hundredth.</p>
<p>Compare the two numbers. Use <code class='latex inline'>\displaystyle > </code> or <code class='latex inline'>\displaystyle < </code> </p><p><code class='latex inline'>\displaystyle \sqrt{63}, 7.5 </code></p>
<p>Suppose <code class='latex inline'>\displaystyle a </code> and <code class='latex inline'>\displaystyle b </code> are positive integers.</p><p>a. Verify that if <code class='latex inline'>\displaystyle a=18 </code> and <code class='latex inline'>\displaystyle b=10 </code>, then <code class='latex inline'>\displaystyle \sqrt{a} \cdot \sqrt{b}=6 \sqrt{5} . </code></p><p>b. Open-Ended Find two other pairs of positive integers <code class='latex inline'>\displaystyle a </code> and <code class='latex inline'>\displaystyle b </code> such that <code class='latex inline'>\displaystyle \sqrt{a} \cdot \sqrt{b}=6 \sqrt{5} . </code></p>
<p>Find each real root.</p><p><code class='latex inline'>\displaystyle -\sqrt[5]{243} </code></p>
<p>Evaluate, to the nearest tenth. </p><p><code class='latex inline'>\displaystyle \sqrt{7.5} </code></p>
<p>Find all the real cube roots of each number. <code class='latex inline'>\displaystyle 0.000343 </code></p>
<p>Write each expression in exponential form.</p><p><code class='latex inline'>\displaystyle \sqrt{-10} </code></p>
<p>Simplify each product.</p><p><code class='latex inline'>\displaystyle -6 \sqrt{15 s^{3}} \cdot 2 \sqrt{75} </code></p>
<p>Convert <code class='latex inline'>\displaystyle \sqrt[10]{1024} </code> into exponential form and evaluate the expression.</p>
<p> Rationalize the denominator.</p><p><code class='latex inline'>\displaystyle \frac{2}{\sqrt[3]{x}} </code></p>
<p>Simplify each radical expression. Use absolute value symbols when needed.</p><p><code class='latex inline'>\displaystyle \sqrt[3]{27 y^{6}} </code></p>
<p>Estimate the square root. Round to the nearest integer.</p><p><code class='latex inline'>\displaystyle \sqrt{242} </code></p>
<p>Tell whether each square root is rational or irrational. Explain.</p><p><code class='latex inline'>\displaystyle \sqrt{0.29} </code></p>
<p>Simplify if possible.</p><p><code class='latex inline'>\displaystyle 5 \sqrt{3}+\sqrt{12} </code></p>
<p>What is the simplified form of <code class='latex inline'>\displaystyle \sqrt{12 y^{5}} </code> ? </p><p><code class='latex inline'>\displaystyle \begin{array}{llll}\text { A } 2 \sqrt{3 y^{5}} & \text { (B) } 4 y^{4} \sqrt{3 y} & \text { (C) } 2 y^{2} \sqrt{3 y}\end{array} </code></p>
<p>Simplify each product.</p><p><code class='latex inline'>\displaystyle -9 \sqrt{28 a^{2}} \cdot \frac{1}{3} \sqrt{63 a} </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle -3 \sqrt{7}+6 \sqrt{3}-8 \sqrt{3}+9 \sqrt{7}-4 \sqrt{7} </code></p>
<p>Find each product or quotient.</p><p><code class='latex inline'>\displaystyle \sqrt[7]{7} \cdot \sqrt[3]{7} </code></p>
<p>Find all the real square roots of each number.</p><p><code class='latex inline'>25</code></p>
<p>Rationalize the denominators and simplify.</p><p><code class='latex inline'>\displaystyle \frac{\sqrt{2}+\sqrt{6}}{\sqrt{1.5}+\sqrt{0.5}} </code></p>
<p>Simplify if possible.</p><p><code class='latex inline'>\displaystyle 4 \sqrt{3}+4 \sqrt[3]{3} </code></p>
<p>Divide and simplify.</p><p><code class='latex inline'>\displaystyle \frac{\sqrt[3]{15x^2}}{\sqrt[3]{5x}} </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle (2 \sqrt{5}-3 \sqrt{2})(\sqrt{5}+2 \sqrt{2}) </code></p>
<p>Evaluate. Expression in rational form(without negative exponents).</p><p><code class='latex inline'>\displaystyle \sqrt[6]{((-2)^3)^2} </code></p>
<p>Rationalize the denominators and simplify.</p><p><code class='latex inline'>\displaystyle \frac{5-\sqrt{21}}{\sqrt{3}-\sqrt{7}} </code></p>
<p>Write the simplest form of <code class='latex inline'>\displaystyle \sqrt[3]{32x^4} </code></p>
<p>Write each expression in radical form, or write each radical in exponential form.</p><p><code class='latex inline'>\displaystyle \sqrt[4]{63} </code></p>
<p>Evaluate. </p><p><code class='latex inline'>\displaystyle \sqrt{1.44} </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle \sqrt[3]{32 a^{5}} </code></p>
<p>Rationalize each numerator.</p><p><code class='latex inline'>\displaystyle \frac{\sqrt{3}}{6 +\sqrt{2}} </code></p>
<p>Vocabulary Determine whether each of the following is a pair of like radicals. If so, add them.</p><p>a. <code class='latex inline'>\displaystyle 3 x \sqrt{11} </code> and <code class='latex inline'>\displaystyle 3 x \sqrt{10} </code></p><p>b. <code class='latex inline'>\displaystyle 2 \sqrt{3 x y} </code> and <code class='latex inline'>\displaystyle 7 \sqrt{3 x y} </code></p><p>c. <code class='latex inline'>\displaystyle 12 \sqrt{13 y} </code> and <code class='latex inline'>\displaystyle 12 \sqrt{6 y} </code></p>
<p>Rationalize each numerator.</p><p><code class='latex inline'>\displaystyle \frac{\sqrt{2}}{5} </code></p>
<p>Rationalize each denominator.</p><p><code class='latex inline'>\displaystyle \frac{5}{\sqrt{7} -4} </code></p>
<p>Convert <code class='latex inline'>16^{\frac{1}{4}}</code> into radical form and evaluate the expression.</p>
<p>Rationalize each denominator. Simplify your answer.</p><p><code class='latex inline'>\displaystyle \frac{3+\sqrt{8}}{2-2 \sqrt{8}} </code></p>
<p>Simplify each expression.</p><p><code class='latex inline'>\displaystyle \sqrt[6]{81 g^{3}} </code></p>
<p>Simplify each expression.</p><p><code class='latex inline'>\displaystyle \frac{x y}{\sqrt[3]{z}} </code></p>
<p>What is the error in</p><p>the problem at the right? What is the</p><p>correct answer?</p><img src="/qimages/56445" /><p><code class='latex inline'>\displaystyle (87 y)^{\frac{2}{3}} </code></p><p><code class='latex inline'>\displaystyle \sqrt[3]{2 x y^{2}} </code></p><p><code class='latex inline'>\displaystyle \beta y^{\frac{2}{3}} </code></p>
<p>Find each square root. Use a calculator if necessary. Round to the nearest hundredth if the result is not a whole number or a simple fraction.</p><p><code class='latex inline'>\sqrt{8100}</code></p>
<p>Find all the real cube roots of each number.</p><p><code class='latex inline'>\displaystyle 0.125 </code></p>
<p>Rationalize the denominators and simplify.</p><p><code class='latex inline'>\displaystyle \frac{4+\sqrt{27}}{2-3 \sqrt{27}} </code></p>
<p>Find each real root.</p><p><code class='latex inline'>\displaystyle \sqrt{36} </code></p>
<p>Is each equation always, sometimes, or never true? Explain your answer.</p><p><code class='latex inline'>\displaystyle \sqrt[3]{x^{3}}=|x| </code></p>
<p>Find each product or quotient.</p><p><code class='latex inline'>\displaystyle \frac{\sqrt[9]{y^{3}}}{\sqrt[3]{y^{9}}} </code></p>
<p>Divide and simplify.</p><p><code class='latex inline'>\displaystyle \frac{\sqrt[3]{250 x^{7} y^{3}}}{\sqrt[3]{2 x^{2} y}} </code></p>
<p>Find all the real fourth roots of each number.</p><p><code class='latex inline'>\frac{10,000}{81}</code></p>
<p>Simplify each expression.</p><p><code class='latex inline'>\displaystyle \sqrt{11}(\sqrt[4]{11}) </code></p>
<p>For what values of <code class='latex inline'>x</code> is <code class='latex inline'>\sqrt{-4x^3}</code> real? Justify your reasoning.</p>
<p>Divide and simplify.</p><p><code class='latex inline'>\displaystyle \frac{\sqrt{21x^{10}}}{\sqrt{7x^5}} </code></p>
<p>Multiply.</p><p><code class='latex inline'>\displaystyle (2+\sqrt{7})(1+3 \sqrt{7}) </code></p>
<p>Simplify each expression.</p><p><code class='latex inline'>\displaystyle \sqrt{3^{2}+4^{2}} </code></p>
<p>Simplify each expression.</p><p><code class='latex inline'>\displaystyle \sqrt[4]{25 x^{2}} </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle 14 \sqrt{20}-3 \sqrt{125} </code></p>
<p>For what values of <code class='latex inline'>\displaystyle a </code> and <code class='latex inline'>\displaystyle b </code> does <code class='latex inline'>\displaystyle \sqrt{a}+\sqrt{b}=\sqrt{a+b} </code> ?</p>
<p>Explain why <code class='latex inline'>\sqrt{a + b } \neq \sqrt{a} + \sqrt{b}</code> for <code class='latex inline'>a > 0</code> and <code class='latex inline'>b> 0</code>.</p>
<p>Multiply each pair of conjugates.</p><p><code class='latex inline'>\displaystyle (\sqrt{3}+\sqrt{5})(\sqrt{3}-\sqrt{5}) </code></p>
<p>Simplify each expression.</p><p><code class='latex inline'>\displaystyle \frac{\sqrt[3]{x}}{\sqrt[6]{x^{5}}} </code></p>
<p>Multiply, if possible. Then simplify.</p><p><code class='latex inline'>\displaystyle \sqrt{-5}\sqrt{5} </code></p>
<p>Evaluate, to the nearest tenth. </p><p><code class='latex inline'>\displaystyle \sqrt{405} </code></p>
<p>Compare the two numbers. Use <code class='latex inline'>\displaystyle > </code> or <code class='latex inline'>\displaystyle < </code> </p><p><code class='latex inline'>\displaystyle 5, \sqrt{22} </code></p>
<p>Simplify each radical expression.</p><p><code class='latex inline'>\displaystyle -5 \sqrt{\frac{162 t^{3}}{2 t}} </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle 3\sqrt{28} - 2\sqrt{50} + \sqrt{63} -3\sqrt{18} </code></p>
<p>Simplify. <code class='latex inline'>\displaystyle \sqrt[3]{-250 x^{6} y^{5}} </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle (\sqrt{1.25}-\sqrt{1.8})(\sqrt{5}+\sqrt{0.2}) </code></p>
<p>Convert <code class='latex inline'>100^{\frac{1}{2}}</code> into radical form and evaluate the expression.</p>
<p>Evaluate. Express answers to three decimals.</p><p><code class='latex inline'>\displaystyle (\sqrt[5]{-1000})^3 </code></p>
<p>Simplify each radical expression. Use absolute value symbols when needed.</p><p><code class='latex inline'>\displaystyle \sqrt[3]{\frac{8}{216}} </code></p>
<p>Simplify each radical expression.</p><p><code class='latex inline'>\displaystyle \sqrt{9 b^{2}} </code></p>
<p>Grades In many classes, a passing test grade is 70 . Using the formula <code class='latex inline'>\displaystyle A=10 \sqrt{R} </code>, what raw score would a student need to get a passing grade after her score is adjusted?</p>
<p>Compare the two numbers. Use <code class='latex inline'>\displaystyle > </code> or <code class='latex inline'>\displaystyle < </code> </p><p><code class='latex inline'>\displaystyle 16, \sqrt{16} </code></p>
<p>(a) What is the formula for the volume of a cylinder?</p><p>(b) Rewrite this formula to express the radius as a function of the volume and height of the cylinder.</p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle 4 \sqrt{216 y^{2}}+3 \sqrt{54 y^{2}} </code></p>
<p>Write the Surface Area as a function of Volume.</p>
<p>Order the numbers from least to greatest.</p><p><code class='latex inline'>\displaystyle \sqrt{2},-20,0.2, \frac{1}{2} </code></p>
<p>Simplify each expression.</p><p><code class='latex inline'>\displaystyle \sqrt{5^{2}+(-12)^{2}} </code></p>
<p>Divide and simplify.</p><p><code class='latex inline'>\displaystyle \frac{\sqrt{56 x^{5} y^{5}}}{\sqrt{7 x y}} </code></p>
<p>Simplify if possible.</p><p><code class='latex inline'>\displaystyle 3 \sqrt{2}+4 \sqrt[3]{2} </code></p>
<p>Evaluate. Express answers in rational form.</p><p><code class='latex inline'>\displaystyle \sqrt[4]{(-9)^{-2}} </code></p>
<p> Express the following with rational denominators.</p><p><code class='latex inline'> \displaystyle \frac{ \sqrt{2} }{\sqrt{2} + \sqrt{3} - \sqrt{5}} </code></p>
<p>Evaluate. Express answers in rational form.</p><p><code class='latex inline'>\displaystyle \sqrt{\frac{100}{121}} </code></p>
<p>Evaluate</p><p><code class='latex inline'>\displaystyle \sqrt[3]{-0.027^4} </code></p>
<p>Find each product or quotient.</p><p><code class='latex inline'>\displaystyle \sqrt[4]{18} \cdot \sqrt{12} </code></p>
<p>Evaluate. Expression in rational form(without negative exponents).</p><p><code class='latex inline'>\displaystyle (\sqrt[3]{-27})^4 </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle \sqrt{12} + 2\sqrt{48}-5\sqrt{27} </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle (3\sqrt{5} + 2\sqrt{10})(-2\sqrt{5} + 5\sqrt{10}) </code></p>
<p>Evaluate.</p><p><code class='latex inline'>\displaystyle \sqrt{36} </code></p>
<p>How many square roots of integers are in the interval between 24 and 25 ?</p>
<p>Simplify each radical expression. Use absolute value symbols when needed.</p><p><code class='latex inline'>\displaystyle \sqrt[5]{32 y^{10}} </code></p>
<p>Rationalize each denominator.</p><p><code class='latex inline'>\displaystyle \frac{5 +\sqrt{2}}{\sqrt{3}} </code></p>
<p>Find each real root.</p><p><code class='latex inline'>\displaystyle \sqrt[4]{810,000} </code></p>
<p>Simplify each expression.</p><p><code class='latex inline'>\displaystyle \sqrt[4]{\sqrt{256}} </code></p>
<p><code class='latex inline'>\displaystyle \begin{array}{l}\text { Geometry The equation } r=\sqrt{\frac{A}{\pi}} \text { gives the radius } r \text { of a circle with area } A \text { . What is } \ \text { the radius of a circle with the given area? Write your answer as a simplified radical } \ \text { and as a decimal rounded to the nearest hundredth. } \ \text { a. } 50 \mathrm{ft}^{2} & \text { b. } 32 \text { in. }^{2} & \text { c. } 10 \mathrm{~m}^{2}\end{array} </code></p>
<p>Write each expression in radical form, or write each radical in exponential form.</p><p><code class='latex inline'>\displaystyle \sqrt[3]{5 x y^{2}} </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle 5 \sqrt{32 x}+4 \sqrt{98 x} </code></p>
<p>For what positive integers <code class='latex inline'>\displaystyle n </code> is each of the statements true?</p><p>a. If <code class='latex inline'>\displaystyle x^{n}=b </code>, then <code class='latex inline'>\displaystyle x </code> is an <code class='latex inline'>\displaystyle n </code> th root of <code class='latex inline'>\displaystyle b </code>.</p><p>b. If <code class='latex inline'>\displaystyle x^{n}=b </code>, then <code class='latex inline'>\displaystyle x=\sqrt[n]{b} </code></p>
<p>Simplify each expression.</p><p><code class='latex inline'>\displaystyle \sqrt[4]{64} </code></p>
<p>Simplify each expression.</p><p><code class='latex inline'>\displaystyle \frac{a b}{\sqrt{c}} </code></p>
<p>Write each expression in exponential form.</p><p><code class='latex inline'>\displaystyle \sqrt{7 x^{3}} </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle 5\sqrt{98} </code></p>
<p>Estimate the square root. Round to the nearest integer.</p><p><code class='latex inline'>\displaystyle \sqrt{61} </code></p>
<p>Convert <code class='latex inline'>\sqrt{121}</code> into exponential form and evaluate the expression.</p>
<p>Express each quotient as a product by using negative exponents.</p><p><code class='latex inline'>\displaystyle \frac{(x + 3)^2}{\sqrt[3]{1- 7x^2}} </code></p>
<p>Compare the two numbers. Use <code class='latex inline'>\displaystyle > </code> or <code class='latex inline'>\displaystyle < </code> </p><p><code class='latex inline'>\displaystyle -\sqrt{38}, 6 </code></p>
<p>Evaluate, to the nearest tenth. </p><p><code class='latex inline'>\displaystyle \sqrt{23} </code></p>
<p>Determine whether each expression is always, sometimes, or never a real number. Assume that <code class='latex inline'>\displaystyle x </code> can be any real number.</p><p><code class='latex inline'>\displaystyle \sqrt{-x} </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle -9\sqrt{12} </code></p>
<p>Evaluate. </p><p><code class='latex inline'>\displaystyle \sqrt{81} </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle 8 \sqrt{6}(3 \sqrt{2}-4 \sqrt{3}-2 \sqrt{6}) </code></p>
<p>Simplify each radical expression if <code class='latex inline'>\displaystyle n </code> is even, and then if <code class='latex inline'>\displaystyle n </code> is odd. <code class='latex inline'>\displaystyle \sqrt[n]{m^{2 n}} </code></p>
<p>Estimate the square root. Round to the nearest integer.</p><p><code class='latex inline'>\displaystyle \sqrt{17} </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle (\sqrt{y}+\sqrt{2})(\sqrt{y}-7 \sqrt{2}) </code></p>
<p>Simplify each radical expression. </p><p><code class='latex inline'>\displaystyle \sqrt{16 b^{5}} </code></p>
<p>Simplify each expression.</p><p><code class='latex inline'>\displaystyle \frac{\sqrt[4]{27}}{\sqrt[4]{3}} </code></p>
<p>Think about a Plan You want to simplify the expression <code class='latex inline'>\displaystyle 4 x^{\frac{3}{2}}+3 \sqrt{x^{3}} </code>.</p> <ul> <li><p>How can you write the radical expression using a rational exponent?</p></li> <li><p>Can you add the resulting terms?</p></li> <li><p>What is the result in simplest form?</p></li> <li><p>Can you write the result in two equivalent forms?</p></li> </ul>
<p>Simplify each radical expression. Use absolute value symbols when needed.</p><p><code class='latex inline'>\displaystyle \sqrt[4]{0.0016} </code></p>
<p>Find all the real fourth roots of each number.</p><p><code class='latex inline'>-16</code></p>
<p> Express each term as a power with a negative exponent.</p><p>a) <code class='latex inline'>\displaystyle \frac{1}{x} </code></p><p>b) <code class='latex inline'>\displaystyle -\frac{2}{x^4} </code></p><p>c) <code class='latex inline'>\displaystyle \frac{1}{\sqrt{x}} </code></p><p>d) <code class='latex inline'>\displaystyle \frac{1}{(\sqrt[3]{x})^2} </code></p>
<p>Rationalize the denominator of each expression.</p><p><code class='latex inline'>\displaystyle \frac{\sqrt{5 x^{4} y}}{\sqrt{2 x^{2} v^{3}}} </code></p>
<p>Simplify if possible.</p><p><code class='latex inline'>\displaystyle 14 \sqrt{x}+3 \sqrt{y} </code></p>
<p>Find each product or quotient.</p><p><code class='latex inline'>\displaystyle \frac{\sqrt{x^{4} y}}{\sqrt[4]{x^{2} y^{8}}} </code></p>
<p>a. Show that <code class='latex inline'>\displaystyle \sqrt{x^{2}}=x </code> by rewriting <code class='latex inline'>\displaystyle \sqrt{x^{2}} </code> in exponential form.</p><p>b. Show that <code class='latex inline'>\displaystyle \sqrt[4]{x^{2}}=\sqrt{x} </code> by rewriting <code class='latex inline'>\displaystyle \sqrt[4]{x^{2}} </code> in exponential form.</p>
<p>What is the conjugate of each expression?</p><p>a. <code class='latex inline'>\displaystyle \sqrt{13}-2 \quad </code> b. <code class='latex inline'>\displaystyle \sqrt{6}+\sqrt{3} \quad </code> c. <code class='latex inline'>\displaystyle \sqrt{5}-\sqrt{10} </code></p>
<p>Simplify each radical expression if <code class='latex inline'>\displaystyle n </code> is even, and then if <code class='latex inline'>\displaystyle n </code> is odd.</p><p><code class='latex inline'>\displaystyle \sqrt{m^{3 n}} </code></p>
<p>Rationalize each denominator.</p><p><code class='latex inline'>\displaystyle \frac{2\sqrt{3}+ 4}{\sqrt{3}} </code></p>
<p>Rationalize the denominators and simplify.</p><p><code class='latex inline'>\displaystyle \frac{4+\sqrt[3]{2}}{\sqrt[3]{2}} </code></p>
<p>Evaluate</p><p><code class='latex inline'>\displaystyle \sqrt[4]{(0.0016)^3} </code></p>
<p>Compare the two numbers. Use <code class='latex inline'>\displaystyle > </code> or <code class='latex inline'>\displaystyle < </code> </p><p><code class='latex inline'>\displaystyle -8, \sqrt{70} </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle \sqrt{7} \times \sqrt{14} </code></p>
<p>Simplify each expression.</p><p><code class='latex inline'>\displaystyle \sqrt{\sqrt{81}} </code></p>
<p>Divide and simplify.</p><p><code class='latex inline'>\displaystyle \frac{\sqrt{20 a b}}{\sqrt{45 a^{2} b^{3}}} </code></p>
<p>Write each expression in exponential form.</p><p><code class='latex inline'>\displaystyle \sqrt[3]{(5 x y)^{6}} </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle 3\sqrt{5} \times 2\sqrt{15} </code></p>
<p>Write each expression in exponential form.</p><p><code class='latex inline'>\displaystyle \sqrt{(7 x)^{3}} </code></p>
<p>Find each real root.</p><p><code class='latex inline'>\displaystyle \sqrt[4]{\frac{1}{160,000}} </code></p>
<p>Compare the two numbers. Use <code class='latex inline'>\displaystyle > </code> or <code class='latex inline'>\displaystyle < </code> </p><p><code class='latex inline'>\displaystyle \sqrt{5}, \sqrt{7} </code></p>
<p>Find each product or quotient.</p><p><code class='latex inline'>\displaystyle \frac{\sqrt{6}}{\sqrt[3]{36}} </code></p>
<p>Simplify each radical expression. Use absolute value symbols when needed.</p><p><code class='latex inline'>\displaystyle \sqrt[4]{x^{20} y^{28}} </code></p>
<p>Evaluate. Express answers in rational form.</p><p><code class='latex inline'>\displaystyle \frac{-\sqrt[3]{512}}{\sqrt[5]{-1024}} </code></p>
<p>Simplify each radical expression.</p><p><code class='latex inline'>\displaystyle \frac{8 \sqrt{7 s}}{\sqrt{28 s^{3}}} </code></p>
<p>Reasoning The diagram at the right shows the dimensions of a kite. The length of the vertical blue crosspiece is <code class='latex inline'>\displaystyle s </code>. What is the length of the horizontal red crosspiece in terms of <code class='latex inline'>\displaystyle s ? </code></p><img src="/qimages/60597" />
<p>Write each expression in exponential form.</p><p><code class='latex inline'>\displaystyle \sqrt[3]{a^{2}} </code></p>
<p>Rationalize each numerator.</p><p><code class='latex inline'>\displaystyle \frac{\sqrt{7}-4}{5} </code></p>
<p>Multiply.</p><p><code class='latex inline'>\displaystyle (2 \sqrt{5}+3 \sqrt{2})^{2} </code></p>
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