**Middle School Number Sense Lesson 79: Exponents with Variables**

Last week we looked at exponents with fractions (in

**ExponentFrac**). Today we will expand our understanding of exponents and work with variables. This concept appeared

**10 times**last year, with a median placement at

**question # 78**.

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To understand how to solve these problems, we first need to understand some of the rules of exponents. It might be helpful to review the

**Logarithm**lesson. Here are some of the rules:

- Adding one to the exponent requires multiplying by the base.
- Subtracting one to the exponent requires dividing by the base.
- Doubling the exponent requires squaring the result.
- Halving the exponent requires taking the square root of the result.

__:__

**Note***Do not be tempted to*

**solve for x**with these problems. In most cases, it is impossible to do mentally.**Example 1:**

**If 8**

^{x}= 12, then 8^{x+1}= ___1. We are adding 1 to the exponent, so we need to multiply by the base.

2. 12 x 8 =

**96**.

**Example 2:**

**If 7**

^{x}= 25, then 7^{x+2}= ___1. We are adding 2 to the exponent, so we need to multiply by the base twice (or by the base squared).

2. 25 x 7 x 7 = 25 x 49 =

**1225**.

**Example 3:**

**If a**

^{3}= 7, then a^{6}= ___1. We are doubling the exponent, so we need to square the result.

2. 7

^{2}=

**49**.

**Example 4:**

**If 10**

^{x}= 9, then 10^{1/2 x}= ___1. We are halving the exponent, so we need to take the square root of the result.

2. The square root of 9 is

**3**.

**Example 5:**

**If 8**

^{x}= 12 and 5^{x}= 7, then 40^{x}= ___1. Since 8 x 5 = 40, and the exponents stay the same, we can simply multiply the results.

2. 12 x 7 =

**84**.

**Example 6:**

**If 2**

^{x}= 3 and 5^{x}= 7, then 20^{x}= ___1. Since 20 =

**2 x 2 x 5**, and the exponents stay the same, we can multiply two of the 2

^{x}results by one 5

^{x}result.

2. 3 x 3 x 7 =

**63**.

**Check back soon for a free worksheet to help you practice ExponentVar.**

**Up Next for Middle School: SolveX2var**