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# 1.6: Evaluate Numerical and Variable Expressions Involving Powers

Difficulty Level: Basic Created by: CK-12
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Practice Algebra Expressions with Exponents
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Do you know how to evaluate a numerical expression when it has powers in it? Casey is having a difficult time doing exactly that. When Casey arrived home from school he looked at his homework. He immediately noticed this problem.

$5^4 + (-2)^4 + 12$

Casey isn't sure how to evaluate this expression. Do you know how to do it? This Concept will show you how to evaluate numerical expressions involving powers. Then you will be able to help Casey at the end of the Concept.

### Guidance

Did you know that you can evaluate numerical and variable expressions involving powers? First, let's identify a numerical and a variable expression.

A numerical expression is a group of numbers and operations that represent a quantity, there isn’t an equal sign.

A variable expression is a group of numbers, operations and variables that represents a quantity, there isn’t an equal sign.

We can combine the order of operations, numerical expressions and variable expressions together with powers.

A power is a number with an exponent and a base . An exponent is a little number that shows the number of times a base is multiplied by itself. The base is the regular sized number that is being worked with.

#### Example A

$6^3 + 5^2 + 25$

Solution: $266$

#### Example B

$16(12^3)$

Solution: $27,648$

#### Example C

$6^2 + 5^3 + 15 - 11$

Solution: $165$

Now let's go back to the dilemma at the beginning of the Concept. Here is the problem that was puzzling to Casey.

$5^4 + (-2)^4 + 12$

First, Casey will need to evaluate the powers.

$5^4 = (5)(5)(5)(5) = 625$

$(-2)^4 = (-2)(-2)(-2)(-2) = 16$

Now we can substitute these values back into the expression.

$625 + 16 + 12 = 653$

This is the answer to Casey's problem.

### Vocabulary

Numerical Expression
A numerical expression is a group of numbers and operations used to represent a quantity.
Variable Expression
A variable expression is a mathematical phrase that contains at least one variable or unknown quantity.
Power
The "power" refers to the value of the exponent. For example, $3^4$ is "three to the fourth power".
Base
When a value is raised to a power, the value is referred to as the base, and the power is called the exponent. In the expression $32^4$ , 32 is the base, and 4 is the exponent.
Exponent
Exponents are used to describe the number of times that a term is multiplied by itself.

### Guided Practice

Here is one for you to try on your own.

Evaluate the expression $5b^4 + 17$ . Let $b=5$ .

Solution

Step 1: Substitute 5 for “ $b$ .”

$5(5)^4 + 17$

Step 2: Simplify the powers.

$& 5(5 \cdot 5 \cdot 5 \cdot 5) + 17\\& 5(625) + 17$

Step 3: Multiply then add to solve.

$& 5(625) + 17\\& 3,125 + 17 = 3,142$

### Explore More

Directions: Evaluate each power.

1. $3^3$
2. $4^2$
3. $(-2)^4$
4. $(-8)^2$
5. $5^3$
6. $2^6$
7. $(-9)^2$
8. $(-2)^6$

Directions: Evaluate each numerical expression.

1. $6^2 + 22$
2. $(-3)^3 + 18$
3. $2^3 + 16 - 4$
4. $(-5)^2 - 19$
5. $(-7)^2 + 52 - 2$
6. $18 + 9^2 - 3$
7. $22 - 3^3 + 7$

Directions: Evaluate each variable expression using the given values.

1. $6a + 4^2 - 2$ , when $a = 3$
2. $a^3 + 14$ , when $a = 6$
3. $2a^2 - 16$ , when $a = 4$
4. $5b^3 + 12$ , when $b = -2$
5. $2x^2 + 52$ , when $x = 4$

Basic

Dec 19, 2012

Feb 26, 2015