# 1.6: Evaluate Numerical and Variable Expressions Involving Powers

**Basic**Created by: CK-12

**Practice**Algebra Expressions with Exponents

Casey missed three days of school this week because of a nasty cold. When he returned to school on Thursday he asked his math teacher what he had missed. Miss Brown wrote a problem on a sheet of paper and handed it to Casey. When he arrived home from school Casey looked at his homework which was the following problem:

Casey didn’t know what to do with

In this concept, you will learn how to evaluate numerical expressions involving powers.

### Evaluating Expressions Involving Powers

A **numerical expression** is an expression made up only of numbers and operations but an expression written with a variable in it, is called a **variable expression**.

Both a numerical expression and a variable expression can include powers.

A **power** is the result of multiplying a number by itself one or more times. The number 16 is the fourth power of 2.

**base** and the number expressing the power is called the **exponent**. The exponent tells you how many times to multiply the **base** times itself.

Let’s look at an example.

Evaluate the following expression:

First, expand the powers to see the operations you need to perform.

Next, apply the order to operations (PEMDAS) to evaluate the expanded expression.

Then, in order from left to right, multiply:

Then, multiply:

Next, from left to right, add:

Then, add:

The answer is 266.

Let’s look at another example.

First, expand the powers to see the operations you need to perform.

Next, write the expression in expanded form.

Next, apply the order to operations (PEMDAS) to evaluate the expanded expression.

Then, in order from left to right multiply:

Then, multiply:

Next, from left to right, add:

Then, add:

Next, subtract:

The answer is 67.

### Examples

#### Example 1

Earlier, you were given a problem about Casey and his confusing homework.

Casey needs to expand the powers so he can see what operations he has to do to evaluate the expression.

First, evaluate the two powers given in the problem.

Next, replace the powers in the expression with their values.

The answer is 653.

#### Example 2

Evaluate the following variable expression when

First, substitute the value

Next, expand the power.

Then, perform the operation in the parenthesis.

Multiply:

Next, multiply:

Then, subtract.

The answer is 116.

#### Example 3

Evaluate the following numerical expression.

First, expand the powers.

Next, write the expression in expanded form.

Then, in order from left to right multiply:

Next, multiply:

Then, in order from left to write, subtract:

\begin{align*}4+1+27\end{align*}

Next, add: \begin{align*}4+1=5\end{align*}

\begin{align*}5+27\end{align*}

Then add: \begin{align*}5+27=32\end{align*}

The answer is 32.

#### Example 4

Evaluate the following variable expression when \begin{align*}m=3\end{align*}

\begin{align*}n^5+3m^2-15\end{align*}

First, substitute \begin{align*}m=3\end{align*}

\begin{align*}2^5+3(3)^2-15\end{align*}

Then, write the expression in expanded form.

\begin{align*}2 \times 2 \times 2 \times 2 \times 2 +3 (3 \times 3)-15\end{align*}

Then, perform the operation in the parenthesis.

First, multiply: \begin{align*}(3 \times 3)=(9)\end{align*}

\begin{align*}2 \times 2 \times 2 \times 2 \times 2 + 3(9)-15\end{align*}

Next, multiply: \begin{align*}3(9)=27\end{align*}

\begin{align*}2 \times 2 \times 2 \times 2 \times 2 +27-15\end{align*}

Next, multiply: \begin{align*}2 \times 2 = 4 \times 2 = 8 \times 2 =16 \times 2 =32
\end{align*}

\begin{align*}32+27-15\end{align*}

Next, add: \begin{align*}32+27=59\end{align*}

\begin{align*}59-15\end{align*}

The answer is 44.

### Review

Expand and evaluate each power.

1. \begin{align*}3^3\end{align*}

2. \begin{align*}4^2\end{align*}

3. \begin{align*}(-2)^4\end{align*}

4. \begin{align*}(-8)^2\end{align*}

5. \begin{align*}5^3\end{align*}

6. \begin{align*}2^6\end{align*}

7. \begin{align*}(-9)^2\end{align*}

8. \begin{align*}(-2)^6\end{align*}

Evaluate each numerical expression. Remember to apply PEMDAS to evaluate the expression accurately.

9. \begin{align*}6^2+22\end{align*}

10. \begin{align*}(-3)^3+18\end{align*}

11. \begin{align*}2^3+16-4\end{align*}

12. \begin{align*}(-5)^2-19\end{align*}

13. \begin{align*}(-7)^2+52-2\end{align*}

14. \begin{align*}18+9^2-3\end{align*}

15. \begin{align*}22-3^3+7\end{align*}

Evaluate each variable expression using the given values.

16. \begin{align*}6a+4^2-2\end{align*}

17. \begin{align*}a^3+14\end{align*}

18. \begin{align*}2a^2-16\end{align*}

19. \begin{align*}5b^3+12\end{align*}

20. \begin{align*}2x^2+52\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.6.

### Resources

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Color | Highlighted Text | Notes | |
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Term | Definition |
---|---|

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 is the base, and 4 is the exponent. |

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

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... |

Numerical expression |
A numerical expression is a group of numbers and operations used to represent a quantity. |

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

Power |
The "power" refers to the value of the exponent. For example, is "three to the fourth power". |

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

Variable Expression |
A variable expression is a mathematical phrase that contains at least one variable or unknown quantity. |

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

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In this concept, you will learn how to evaluate numerical expressions involving powers.