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Solutions
57 Videos

If `x > 1`

, which is greater, `x^{-2}`

or `x^2`

? Why?

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Q1a

Are there values of `x`

that make the statement `x^{-2} > x^2`

true? Explain.

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Q1b

Write each as a single power. Then evaluate. Express answers in rational form.

```
\displaystyle
(-7)^3(-7)^{-4}
```

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Q2a

Write each as a single power. Then evaluate. Express answers in rational form.

```
\displaystyle
\frac{(-2)^8}{(-2)^3}
```

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0.21mins

Q2b

Write each as a single power. Then evaluate. Express answers in rational form.

```
\displaystyle
\frac{(5)^{-3}(5)^6}{5^3}
```

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Q2c

Write each as a single power. Then evaluate. Express answers in rational form.

```
\displaystyle
\frac{4^{-10}(4^{-3})^6}{(4^{-4})^8}
```

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0.41mins

Q2d

Write each as a single power. Then evaluate. Express answers in rational form.

```
\displaystyle
(11)^{9}(\frac{1}{11})^{7}
```

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Q2e

Write each as a single power. Then evaluate. Express answers in rational form.

```
\displaystyle
(\frac{(-3)^7(-3)^4}{(-3^4)^3})^{-3}
```

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Q2f

Express each radical in exponential form and each power in radical form.

```
\displaystyle
\sqrt[3]{x^7}
```

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Q3a

Express each radical in exponential form and each power in radical form.

```
\displaystyle
y^{\frac{8}{5}}
```

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Q3b

Express each radical in exponential form and each power in radical form.

```
\displaystyle
(\sqrt{p})^{11}
```

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Q3c

Express each radical in exponential form and each power in radical form.

```
\displaystyle
m^{1.25}
```

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Q3d

Evaluate. Expression in rational form(without negative exponents).

```
\displaystyle
(\frac{2}{5})^{-3}
```

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0.25mins

Q4a

Evaluate. Expression in rational form(without negative exponents).

```
\displaystyle
(\frac{16}{225})^{-0.5}
```

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Q4b

Evaluate. Expression in rational form(without negative exponents).

```
\displaystyle
(\frac{16}{225})^{-0.5}
```

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Q4c

Evaluate. Expression in rational form(without negative exponents).

```
\displaystyle
(\sqrt[3]{-27})^4
```

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Q4d

Evaluate. Expression in rational form(without negative exponents).

```
\displaystyle
\sqrt[5]{-32}(\sqrt[6]{64})^5
```

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0.55mins

Q4e

Evaluate. Expression in rational form(without negative exponents).

```
\displaystyle
\sqrt[6]{((-2)^3)^2}
```

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1.36mins

Q4f

Evaluate. Expression in rational form(without negative exponents).

```
\displaystyle
a^{\frac{3}{2}}(a^{- \frac{3}{2}})
```

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0.17mins

Q5a

Evaluate. Expression in rational form(without negative exponents).

```
\displaystyle
\frac{b^{0.8}}{b^{-0.2}}
```

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0.14mins

Q5b

Evaluate. Expression in rational form(without negative exponents).

```
\displaystyle
\frac{c(c^{\frac{5}{6}})}{c^2}
```

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Q5c

Evaluate. Expression in rational form(without negative exponents).

```
\displaystyle
\frac{d^{-5}d^{\frac{11}{2}}}{(d^{-3})^2}
```

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Q5d

Evaluate. Expression in rational form(without negative exponents).

```
\displaystyle
((e^{-2})^{\frac{7}{2}})^{-2}
```

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0.18mins

Q5e

Evaluate. Expression in rational form(without negative exponents).

```
\displaystyle
((f^{- \frac{1}{6}})^{\frac{6}{5}})^{-1}
```

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Q5f

Explain why `\sqrt{a + b } \neq \sqrt{a} + \sqrt{b}`

for `a > 0`

and `b> 0`

.

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0.55mins

Q6

Evaluate the expression for the given values. Express answers in rational form.

```
\displaystyle
(5x)^2(2x)^3
```

where `x = -2`

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0.42mins

Q7a

Evaluate the expression for the given values. Express answers in rational form.

```
\displaystyle
\frac{8m^{-5}}{(2m)^{-3}}
```

where `x = 4`

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0.35mins

Q7b

Evaluate the expression for the given values. Express answers in rational form.

```
\displaystyle
\frac{2w(3w^{-2})}{(2w)^2}
```

where `w= -3`

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1.00mins

Q7c

Evaluate the expression for the given values. Express answers in rational form.

```
\displaystyle
\frac{(9y)^2}{(3y^{-1})^3}
```

where `y= -2`

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Q7d

Evaluate the expression for the given values. Express answers in rational form.

```
\displaystyle
(6(x^{-4})^3)^{-1}
```

where `x= -2`

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Q7e

Evaluate the expression for the given values. Express answers in rational form.

```
\displaystyle
\frac{(-2x^{-2})^3(6x)^2}{2(-3x^{-1})^3}
```

where `x= \frac{1}{2}`

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1.43mins

Q7f

Simplify. Write the expression using only positive exponents. All variables are positive.

```
\displaystyle
\sqrt[3]{27x^3 y^9}
```

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0.39mins

Q8a

Simplify. Write the expression using only positive exponents. All variables are positive.

```
\displaystyle
\sqrt{\frac{a^6b^5}{a^8b^3}}
```

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0.35mins

Q8b

Simplify. Write the expression using only positive exponents. All variables are positive.

```
\displaystyle
\frac{m^{\frac{3}{2}}n^{-2}}{m^{\frac{7}{2}}n^{-\frac{3}{2}}}
```

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Q8c

Simplify. Write the expression using only positive exponents. All variables are positive.

```
\displaystyle
\frac{\sqrt[4]{x^{-16}}(x^6)^{-6}}{(x^4)^{- \frac{11}{2}}}
```

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Q8d

Simplify. Write the expression using only positive exponents. All variables are positive.

```
\displaystyle
((-x^{0.5})^3)^{-1.2}
```

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1.31mins

Q8e

Simplify. Write the expression using only positive exponents. All variables are positive.

```
\displaystyle
\frac{\sqrt{x^6(y^3)^{-2}}}{(x^3y)^{-2}}
```

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Q8f

Identify the type of function (linear, quadratic, or exponential) for each table of values.

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Q9a

Identify the type of function (linear, quadratic, or exponential) for each table of values.

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Q9b

Identify the type of function (linear, quadratic, or exponential) for each table of values.

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Q9c

Identify the type of function (linear, quadratic, or exponential) for each table of values.

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0.34mins

Q9d

Identify the type of function (linear, quadratic, or exponential) for each table of values.

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Q9e

Identify the type of function (linear, quadratic, or exponential) for each table of values.

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0.31mins

Q9f

For each exponential function, state the base function, `y = 5^x`

. Then state the transformations that map the base function onto the given function. Use transformations to sketch each graph.

```
\displaystyle
y = (\frac{1}{2})^{\frac{x}{2}} -3
```

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2.11mins

Q11a

For each exponential function, state the base function, `y = 5^x`

. Then state the transformations that map the base function onto the given function. Use transformations to sketch each graph.

```
\displaystyle
y = \frac{1}{4}(x)^{-x} + 1
```

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2.08mins

Q11b

For each exponential function, state the base function, `y = 5^x`

. Then state the transformations that map the base function onto the given function. Use transformations to sketch each graph.

```
\displaystyle
y =-2(3)^{2x +4}
```

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2.33mins

Q11c

`y = 5^x`

. Then state the transformations that map the base function onto the given function. Use transformations to sketch each graph.

```
\displaystyle
y = \frac{-1}{10}(5)^{3x - 9} + 10
```

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3.56mins

Q11d

Find the missing rows:

Functions: `V(t) = 100(1.08)^t`

Exponential Growth or Decay?(Pick one):

Initial Value:

Growth or Decay Rate: %

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0.27mins

Q13a

Find the missing rows:

Functions: `P(n) = 32(0.95)^n`

Exponential Growth or Decay?(Pick one):

Initial Value:

Growth or Decay Rate: %

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0.24mins

Q13b

Find the missing rows:

Functions: `A(x) = 5(3)^x`

Exponential Growth or Decay?(Pick one):

Initial Value:

Growth or Decay Rate: %

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0.20mins

Q13c

Find the missing rows:

Functions: `Q(n) = 600(\frac{5}{8})^n`

Exponential Growth or Decay?(Pick one):

Initial Value:

Growth or Decay Rate: %

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0.25mins

Q13d

A hot cup of coffee cools according to the equation

```
\displaystyle
T(t) = 69(\frac{1}{2})^{\frac{t}{30}} + 21
```

where `T`

is the temperature in degrees Celsius and `t`

is the time in minutes.

a) Which part of the equation indicates that this is an example of exponential decay?

b) What was the initial temperature of the coffee?

c) Use your knowledge of transformations to sketch the graph of this function.

d) Determine the temperature of the coffee, to the nearest degree, after 48 min.

e) Explain how the equation would change if the coffee cooled faster.

f) Explain how the graph would change if the coffee cooled faster.

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2.34mins

Q14

The value of a car after it is purchased depreciates according to the formula

```
\displaystyle
V(n) = 28 000(0.875)^n
```

where `V(n)`

is the car's value in the nth year since it was purchased.

a) What is the purchase price of the car?

b) What is the annual rate of depreciation?

c) What is the car’s value at the end of 3 years? What is its value at the end of30 months?

d) How much value does the car lose in its first year? How much value does it lose in its fifth year?

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3.23mins

Q15

Write the equation that models each situation. In each case, describe each part ofyour equation.

a) the percent of a pond covered by water lilies if they cover one—third of a pond now and each week they increase their coverage by 10%

b) the amount remaining of the radioactive isotope U238 if it has a half—life of `4.5 \times 10^9`

years

c) the intensity of light if each gel used to change the colour of a spotlight reduces the intensity of the light by 4%

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2.03mins

Q16

The population of a city is growing at an average rate of `3\%`

per year. In 1990, the population was 45 000.

a) Write an equation that models the growth of the city. Explain what each part of the equation represents.

b) Use your equation to determine the population of the city in 2007.

c) Determine the year during which the population will have doubled.

d) Suppose the population took only 10 years to double. What growth rate would be required for this to have happened?

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Q17