Use algebra to express each situation. Write an algebraic expression to represent the model.
a) Jeanne ran 4 km.
b) Klaus drove an unknown distance, twice.
c) Evelyn ran 3 km plus an unknown distance.
d) Suki painted her house with two coats of paint.
a) Build a volume model to represent a cube with side length 3 cm. Sketch the model and label the length, width, and height.
b) What is the volume of the cube? Write this as a power.
c) Write an expression for the area of one face of the cube as a power. Evaluate the area of one face.
Evaluate
a) \displaystyle
4^5
b) \displaystyle
(-3)^4
c) \displaystyle
(\frac{2}{5})^3
d) \displaystyle
1.05^8
$100 is put into a bank account that pays interest so that the amount in the account grows according to the expression 100(1.06)n, where n is the number of years. Find the amount in the account after
a) 5 years
b) 10 years
The half-life of carbon-14 (C-14) is 5700 years.
a) Copy and complete the table for a 50-g sample of C-14.
b) Construct a graph of the amount of C-14 remaining versus time, in years. Describe the shape of the graph.
c) Approximately how much C-14 will remain after 20 000 years?
d) How long will it take until only 1 g of C-14 remains?
Write it as a single power.
\displaystyle
2^3 \times 2^2 \times 2^4
Write it as a single power.
\displaystyle
6^7 \div 6^2 \div 6^3
Write it as a single power.
\displaystyle
[(-4)^2]^3
Write it as a single power.
\displaystyle
\frac{7^4 \times 7^5}{(7^4)^2}
Simplify
\displaystyle
\frac{n^5 \times n^3}{n^4}
Simplify
\displaystyle
cd^3 \times c^4 d^2
Simplify
\displaystyle
\frac{2ab^2 \times 3a^3b^3}{(4ab^2)^2}
Identify the coefficient and the variable part of each term.
a) \displaystyle
5y
b) \displaystyle
uv
c) \displaystyle
\frac{1}{2}ab^2
d) \displaystyle
-de^2f
e) \displaystyle
8
f) \displaystyle
16 i^2 -7v^2
Classify each polynomial by the number of terms.
a) \displaystyle
x^2 + 3x - 5
b) \displaystyle
24xy
c) \displaystyle
a + 2b - c + 3
d) \displaystyle
- \frac{2}{3}
e) \displaystyle
16u^2 -7v^2
In a hockey tournament, teams are awarded 3 points for a win, 2 points for an overtime win, and 1 point for an overtime loss.
a) Write an expression that describes the number of points a team has.
b) Use your expression to find the number of points earned by a team that has 4 wins, 1 overtime win, and 2 overtime losses.
State the degree of each term.
a) \displaystyle
3x^2
b) \displaystyle
6n^4
c) \displaystyle
17
d) \displaystyle
abc^2
State the degree of each polynomial.
a) \displaystyle
3y - 5
b) \displaystyle
2d^2 -d
c) \displaystyle
3w -6w^2 + 4
d) \displaystyle
3x^3 -5x^2 + x
Identify the like terms in each set.
a) \displaystyle
2p, 3q, -2, p, 3q^2
Identify the like terms in each set.
a) \displaystyle
5x^2, 5x , x^5, -5x^2, 3x^2
Simplify by collecting like terms.
a) \displaystyle
4x - 3 + 6x + 5
b) \displaystyle
7k + 5m -k - 6m
Simplify by collecting like terms.
\displaystyle
6a^2 - 5a + 3 - 3a^2 + 5a -4
Simplify by collecting like terms.
\displaystyle
3x^2 - 4xy + 5y^2 - 6 + 3x^2+ 4xy -2
Simplify
a) \displaystyle
(4x +3) + (3x - 2)
b) \displaystyle
(5k -2) + (3k - 5)
c) \displaystyle
(6u+1) - (2u + 5)
Simplify
\displaystyle
(y^2 -3y)-(2y^2 - 5y)
Simplify
\displaystyle
(2a^2 -4a -2)-(a^2 -4a + 2)
Simplify
\displaystyle
(3v -2)-(v-3)+(2v - 7)
A rectangular window frame has dimensions expressed by 3x and 2x - 5. Find a simplified expression for its perimeter.
Expand.
a) \displaystyle
3(y - 7)
b) \displaystyle
-2(x + 3)
c) \displaystyle
m(5m -3)
d) \displaystyle
-4k(2k + 6)
Expand.
a) \displaystyle
-5(p^2 + 3p - 1)
b) \displaystyle
4b(b^2 - 2b + 5)
Expand and simplify.
a) \displaystyle
2(q - 5) + 4(3q +2)
b) \displaystyle
5x(2x -4) -3(2x^2 +8)
Expand and simplify.
a) \displaystyle
-3(2m - 6) - (8 - 6m)
b) \displaystyle
4(2d-5) + 3(d^2 -3d) -2d(d+ 1)
Simplify
\displaystyle
2[4 + 3(x - 5)]
Simplify
\displaystyle
-3[9 - 2(k + 3)+ 5k]