**Middle School Number Sense Lesson 78: Exponents with Fractions**

Exponents play a major part in number sense tests, and we have had many lessons which included them. Today we will work on exponents with fractions in them. This concept appeared

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

**question # 78**.

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To understand exponents with fractions, we need to understand both powers and roots. In any fractional exponent:

- the
**numerator**represents the power, and - the
**denominator**represents the root.

**4**to the

**3/2**power, we are really:

- cubing it (or taking it to the 3rd power), and
- taking the square root...though not necessarily in that order.

Just like I have a rule that says

**"divide before you multiply,"**I prefer to

**"take the root before the power."**This ensures I am working with smaller numbers, which are easier to manipulate and calculate than larger ones.

**Example 1:**

**81**

^{1/2}= ___1. Think of this as the square root of (81 to the first power).

2. 81

^{1}= 81.

3. √81 =

**9**.

**Example 2:**

**27**

^{1/3}= ___1. Think of this as the cube root of (27 to the 1st power).

2. 27

^{1}= 27.

3. The cube root of 27 is

**3**.

**Example 3:**

**9**

^{3/2}= ___1. Think of this as the square root of 9

^{3}, or (√9)

^{3}.

2. √9 = 3.

3. 3

^{3}=

**27**.

**Example 4:**

**8**

^{2/3}= ___1. Think of this as the cube root of 8

^{2}, or (the cube root of 8)

^{2}.

2. The cube root of 8 is 2.

3. 2

^{2}=

**4**.

**Example 5:**

**16**

^{1.25}= ___1. Think of 1.25 as 5/4. This means we are taking (the 4th root of 16) to the 5th power.

2. The 4th root of 16 is 2.

3. 2

^{5}=

**32**.

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

**Up Next for Middle School: ExponentVar**