<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />

# 1.3: Algebra Expressions with Exponents

Difficulty Level: At Grade Created by: CK-12
Estimated16 minsto complete
%
Progress
Practice Algebra Expressions with Exponents
Progress
Estimated16 minsto complete
%

What if you knew the volume of a cube was represented by the formula V=s3\begin{align*} V = s^3\end{align*}, where s the length of a side. You measure the cube's side to be 4 inches. How could you find its volume? After completing this Concept, you'll be able to evaluate exponential expressions like this one.

### Watch This

For a more detailed review of exponents and their properties, check out the video at http://www.mathvids.com/lesson/mathhelp/863-exponents---basics.

### Guidance

Many formulas and equations in mathematics contain exponents. Exponents are used as a short-hand notation for repeated multiplication. For example:

22222=22=23\begin{align*} 2 \cdot 2 &= 2^2\\ 2 \cdot 2 \cdot 2 &= 2^3 \end{align*}

The exponent stands for how many times the number is used as a factor (multiplied). When we deal with integers, it is usually easiest to simplify the expression. We simplify:

2223=4=8\begin{align*} 2^2 &= 4\\ 2^3 &= 8\end{align*}

However, we need exponents when we work with variables, because it is much easier to write x8\begin{align*}x^8\end{align*} than xxxxxxxx\begin{align*}x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x\end{align*}.

To evaluate expressions with exponents, substitute the values you are given for each variable and simplify. It is especially important in this case to substitute using parentheses in order to make sure that the simplification is done correctly.

#### Example A

The area of a circle is given by the formula A=πr2\begin{align*}A = \pi r^2\end{align*}. Find the area of a circle with radius r=17 inches\begin{align*}r = 17 \ inches\end{align*}.

Substitute values into the equation.

A=πr2 Substitute 17 for r.A=π(17)2π1717907.9202 Round to 2 decimal places.\begin{align*}& A = \pi r^2 \qquad \ \text{Substitute} \ 17 \ \text{for} \ r.\\ & A = \pi (17)^2 \quad \pi \cdot 17 \cdot 17 \approx 907.9202 \ldots \ \text{Round to} \ 2 \ \text{decimal places.}\end{align*}

The area is approximately 907.92 square inches.

#### Example B

Find the value of x2y3x3+y2\begin{align*}\frac { x^2y^3 } { x^3 + y^2 }\end{align*}, for x=2\begin{align*}x = 2\end{align*} and y=4\begin{align*}y = -4\end{align*}.

Substitute the values of x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*} in the following.

x2y3x3+y2=(2)2(4)3(2)3+(4)2 Substitute 2 for x and 4 for y.4(64)8+16=25624=323Evaluate expressions: (2)2=(2)(2)=4 and (2)3=(2)(2)(2)=8. (4)2=(4)(4)=16 and (4)3=(4)(4)(4)=64.\begin{align*}& \frac { x^2y^3 } { x^3 + y^2 } = \frac { (2)^2 (-4)^3 } { (2)^3 + (-4)^2 } \qquad \ \text{Substitute} \ 2 \ \text{for} \ x \ \text{and} \ -4 \ \text{for} \ y.\\ & \frac { 4(-64) } { 8 + 16 } = \frac { - 256 } { 24 } = \frac{-32}{3} \qquad \text{Evaluate expressions:} \ (2)^2 = (2)(2) = 4 \ \text{and}\\ & \qquad \qquad \qquad \qquad \qquad \qquad \ (2)^3 = (2)(2)(2) = 8. \ (-4)^2 = (-4)(-4) = 16 \ \text{and}\\ & \qquad \qquad \qquad \qquad \qquad \qquad \ (-4)^3 = (-4)(-4)(-4) = -64.\end{align*}

#### Example C

The height (h)\begin{align*}(h)\end{align*} of a ball in flight is given by the formula h=32t2+60t+20\begin{align*}h = - 32t^2 + 60t + 20\end{align*}, where the height is given in feet and the time (t)\begin{align*}(t)\end{align*} is given in seconds. Find the height of the ball at time t=2 seconds\begin{align*}t = 2 \ seconds\end{align*}.

Solution

h=32t2+60t+20=32(2)2+60(2)+20Substitute 2 for t.=32(4)+60(2)+20=12\begin{align*}h &= -32t^2 + 60t + 20\\ &= -32(2)^2 + 60(2) + 20 \qquad \text{Substitute} \ 2 \ \text{for} \ t.\\ &= -32(4) + 60(2) + 20\\ &= 12\end{align*}

The height of the ball is 12 feet.

Watch this video for help with the Examples above.

### Vocabulary

• Exponents are used as a short-hand notation for repeated multiplication. For example:

22222=22=23\begin{align*} 2 \cdot 2 &= 2^2\\ 2 \cdot 2 \cdot 2 &= 2^3 \end{align*}

The exponent stands for how many times the number is used as a factor (multiplied).

### Guided Practice

Find the value of a2+b2a2b2\begin{align*}\frac { a^2+b^2 } { a^2-b^2 }\end{align*}, for a=1\begin{align*}a = -1\end{align*} and 5\begin{align*}5\end{align*}.

Substitute the values of x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*} in the following.

a2+b2a2b2=(1)2+(5)2(1)2(5)2 Substitute 1 for a and 5 for b.1+25125=2624=1312Evaluate and simplify expressions.\begin{align*}& \frac { a^2+b^2 } { a^2-b^2 } = \frac { (-1)^2+(5)^2 } { (-1)^2-(5)^2 } \qquad \ \text{Substitute} \ -1 \ \text{for} \ a \ \text{and} \ 5 \ \text{for} \ b.\\ & \frac { 1+25 } { 1-25 } = \frac { 26 } { 24 }=\frac{13}{12} \qquad \text{Evaluate and simplify expressions.} \end{align*}

### Practice

Evaluate 1-8 using x=1, y=2, z=3,\begin{align*}x = -1, \ y = 2, \ z = -3,\end{align*} and w=4\begin{align*}w = 4\end{align*}.

1. 8x3\begin{align*} 8x^3\end{align*}
2. 5x26z3\begin{align*}\frac { 5x^2 } { 6z^3 } \end{align*}
3. 3z25w2\begin{align*} 3z^2 - 5w^2\end{align*}
4. x2y2\begin{align*} x^2 - y^2 \end{align*}
5. z3+w3z3w3\begin{align*} \frac { z^3 + w^3 } { z^3 - w^3 } \end{align*}
6. 2x33x2+5x4\begin{align*} 2x^3 - 3x^2 + 5x - 4\end{align*}
7. 4w3+3w2w+2\begin{align*} 4w^3 + 3w^2 - w + 2 \end{align*}
8. 3+1z2\begin{align*}3 + \frac{ 1 } { z^2 }\end{align*}

For 9-10, use the fact that the volume of a box without a lid is given by the formula V=4x(10x)2\begin{align*} V = 4x(10 - x)^2\end{align*}, where x\begin{align*}x\end{align*} is a length in inches and V\begin{align*}V\end{align*} is the volume in cubic inches.

1. What is the volume when x=2\begin{align*}x = 2\end{align*}?
2. What is the volume when x=3\begin{align*}x = 3\end{align*}?

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Please to create your own Highlights / Notes
Show More

### Vocabulary Language: English

Evaluate

To evaluate an expression or equation means to perform the included operations, commonly in order to find a specific value.

Exponent

Exponents are used to describe the number of times that a term is multiplied by itself.

Expression

An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols.

Integer

The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...

Parentheses

Parentheses "(" and ")" are used in algebraic expressions as grouping symbols.

substitute

In algebra, to substitute means to replace a variable or term with a specific value.

Volume

Volume is the amount of space inside the bounds of a three-dimensional object.

### Image Attributions

Show Hide Details
Description
Difficulty Level:
At Grade
Authors:
Sources:
Tags:
Subjects:

## Concept Nodes:

Grades:
Date Created:
Aug 13, 2012
Last Modified:
Sep 13, 2016
Files can only be attached to the latest version of Modality
Please wait...
Please wait...
Image Detail
Sizes: Medium | Original

MAT.ALG.132.L.1
Here