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## Course: Algebra 2 > Unit 6

Evaluating fractional exponents.

- Evaluating fractional exponents: negative unit-fraction
- Evaluating fractional exponents: fractional base
- Evaluating quotient of fractional exponents
- Evaluating mixed radicals and exponents

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## Video transcript

## Fractional Exponents

## What are Fractional Exponents?

Look at the figure given below to understand how fractional exponents are represented.

Some examples of fractional exponents that are widely used are given below:

## Fractional Exponents Rules

- Rule 1: a 1/m × a 1/n = a (1/m + 1/n)
- Rule 2: a 1/m ÷ a 1/n = a (1/m - 1/n)
- Rule 3: a 1/m × b 1/m = (ab) 1/m
- Rule 4: a 1/m ÷ b 1/m = (a÷b) 1/m
- Rule 5: a -m/n = (1/a) m/n

## Simplifying Fractional Exponents

## Multiply Fractional Exponents With the Same Base

## How to Divide Fractional Exponents?

The division of fractional exponents can be classified into two types.

- Division of fractional exponents with different powers but the same bases
- Division of fractional exponents with the same powers but different bases

## Negative Fractional Exponents

## Fractional Exponents Examples

Example 1: Evaluate 18 1/2 ÷ 2 1/2 .

Therefore, 2 1/2 × 4 1/4 × 8 1/8 = 2 11/8 .

Example 3: Evaluate 3 2/3 ÷ 9 1/2

Solution: To solve this, we will reduce 9 1/2 to the simplest form. So, we have

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## Fractional Exponents Questions

## FAQs on Fractional Exponents

What do fractional exponents mean.

## What is the Rule for Fractional Exponents?

## What To Do With Negative Fractional Exponents?

## How To Solve Fractional Exponents?

## How To Add Fractional Exponents?

## How To Divide Fractional Exponents?

## Fractional Exponents

Also called "Radicals" or "Rational Exponents"

## Whole Number Exponents

First, let us look at whole number exponents :

The exponent of a number says how many times to use the number in a multiplication.

In this example: 8 2 = 8 × 8 = 64

Another example: 5 3 = 5 × 5 × 5 = 125

But what if the exponent is a fraction?

First, the Laws of Exponents tell us how to handle exponents when we multiply:

## Example: x 2 x 3 = (xx)(xxx) = xxxxx = x 5

Which shows that x 2 x 3 = x (2+3) = x 5

So let us try that with fractional exponents:

## Example: What is 9 ½ × 9 ½ ?

9 ½ × 9 ½ = 9 (½+½) = 9 (1) = 9

## Try Another Fraction

Let us try that again, but with an exponent of one-quarter (1/4):

16 ¼ × 16 ¼ × 16 ¼ × 16 ¼ = 16 (¼+¼+¼+¼) = 16 (1) = 16

So 16 ¼ used 4 times in a multiplication gives 16,

and so 16 ¼ is a 4th root of 16

## General Rule

It worked for ½ , it worked with ¼ , in fact it works generally:

## Example: What is 27 1/3 ?

## What About More Complicated Fractions?

What about a fractional exponent like 4 3/2 ?

That is really saying to do a cube (3) and a square root (1/2), in any order.

A fraction (like m/n ) can be broken into two parts:

So, because m/n = m × (1/n) we can do this:

x m/n = x (m × 1/n) = (x m ) 1/n = n √ x m

The order does not matter, so it also works for m/n = (1/n) × m :

x m/n = x (1/n × m) = (x 1/n ) m = ( n √ x ) m

## Example: What is 4 3/2 ?

4 3/2 = 4 3×(1/2) = √(4 3 ) = √(4×4×4) = √(64) = 8

4 3/2 = 4 (1/2)×3 = (√4) 3 = (2) 3 = 8

Either way gets the same result.

## Example: What is 27 4/3 ?

27 4/3 = 27 4×(1/3) = 3 √ 27 4 = 3 √ 531441 = 81

27 4/3 = 27 (1/3)×4 = ( 3 √ 27 ) 4 = (3) 4 = 81

It was certainly easier the 2nd way!

## Now ... Play With The Graph!

- Start with m=1 and n=1, then slowly increase n so that you can see 1/2, 1/3 and 1/4
- Then try m=2 and slide n up and down to see fractions like 2/3 etc
- Now try to make the exponent −1
- Lastly try increasing m, then reducing n, then reducing m, then increasing n: the curve should go around and around

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## 1.5: Equations with Rational Exponents

## Solving Equations Involving Rational Exponents

Definition: Rational Exponents

Example \(\PageIndex{1}\): Evaluate a Number Raised to a Rational Exponent

Evaluate \({64}^{-\tfrac{1}{3}}\)

- Isolate the expression with the rational exponent
- If the numerator of the reciprocal power is an even number, the solution must be checked because the solution involves the squaring process which can introduce extraneous roots.
- If the denominator of the reciprocal power is an even number, this is equivalent to taking an even root so +/- must be included.

Example \(\PageIndex{2}\): Solve an Equation Containing a Variable Raised to a Rational Exponent

Solve the equation in which a variable is raised to a rational exponent: \(x^{\tfrac{3}{4}} = 8\).

Solve \(x^\tfrac{5}{4}+36=4\).

Solve \((x+5)^\tfrac{2}{3}= 64\)

The solution does not need to be checked! Solution Set: \( \{ 509, -517 \} \)

a. \( \{ 129, -121 \} \qquad \) b. \( \{ -1 \} \qquad \) c. \( \{ \} \)

Example \(\PageIndex{7}\): Solve an Equation involving Rational Exponents and Factoring

Solve \(3x^{\tfrac{3}{4}} = x^{\tfrac{1}{2}}\).

Now we have two factors and can use the zero factor theorem.

\( x^{\tfrac{1}{2}}\left (3x^{\tfrac{1}{4}}-1 \right )= 0 \)

The solution set is \( {\Large\{} 0, \dfrac{1}{81} {\Large\}}\).

## What is a Fractional Exponent?

b n/m = ( m √ b ) n = m √ (b n ), let us define some the terms of this expression.

The radicand is the under the radical sign √. In this case our radicand is b n

This is the number whose root is being calculated. The base is denoted with a letter b.

## How to Solve Fractional Exponents?

Let’s know how to solve fractional exponents with the help of examples below.

= ∛ (27 4 ) = 3 √ (531441) = 81

- Simplify: 125 1/3 125 1/3 = ∛125 = [(5) 3 ] 1/3 = (5) 1 = 5
- Calculate: (8/27) 4/3 (8/27) 4/3 8 = 2 3 and 27 = 3 3 So, (8/27) 4/3 = (2 3 /3 3 ) 4/3 = [(2/3) 3 ] 4/3 = (2/3) 4 = 2/3 × 2/3 × 2/3 × 2/3 = 16/81

x 1/3 × x 1/3 × x 1/3 = x (1/3 + 1/3 + 1/3)

= x 2/3 , this can be expressed as ∛x 2

8 1/3 x 8 1/3 = 8 1/3 + 1/3 = 8 2/3

And since the cube root of 8 can be found easily,

When dividing fractional exponent with the same base, we subtract the exponents. For instance:

16 1/2 ÷ 16 1/4 = 16 (1/2 – 1/4)

You can notice that, 16 1/2 = 4 and 16 1/4 = 2.

In general; if the base x = a/b,

Then, (a/b) -n/m = (b/a) n/m .

## Previous Lesson | Main Page | Next Lesson

## Dealing with fractional exponents

## How fractional exponents are related to roots

The rule for fractional exponents:

???x^{\frac{a}{b}}??? ???=??? ???\sqrt[b]{x^a}???

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## Changing fractional exponents into roots and vice versa

## Take the course

???\sqrt{4 \cdot 4 \cdot 4}???

in a fractional exponent, think of the numerator as an exponent, and the denominator as the root

Another rule for fractional exponents:

???\left[(x)^a\right]^{\frac{1}{b}}???

???\left[(x)^{\frac{1}{b}}\right]^a???

## Let's do another example.

???\left(\frac{1}{9}\right)^{\frac{3}{2}}???

???\left[\left(\frac{1}{9}\right)^{\frac{1}{2}}\right]^3???

???\left[\sqrt{\frac{1}{9}}\right]^3???

???\left(\frac{\sqrt{1}}{\sqrt{9}}\right)^3???

???\left(\frac{1}{3}\right)^3???

???\left(\frac{1}{3}\right)\left(\frac{1}{3}\right)\left(\frac{1}{3}\right)???

## How to get rid of fractional exponents

Write the expression without fractional exponents.

???\left(\frac{1}{6}\right)^{\frac{3}{2}}???

We can rewrite the expression by breaking up the exponent.

???\left[\left(\frac{1}{6}\right)^3\right]^{\frac{1}{2}}???

???\left(\frac{1}{6} \cdot \frac{1}{6} \cdot \frac{1}{6}\right)^{\frac{1}{2}}???

???\sqrt{\frac{1}{6} \cdot \frac{1}{6} \cdot \frac{1}{6}}???

???\frac{\sqrt{1}}{\sqrt{216}}???

???\frac{1}{\sqrt{36 \cdot 6}}???

???\frac{1}{\sqrt{36} \sqrt{6}}???

We need to rationalize the denominator.

???\frac{1}{6\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}???

???\frac{\sqrt{6}}{6 \cdot 6}???

## What happens if you have a negative fractional exponent?

You should deal with the negative sign first, then use the rule for the fractional exponent.

???\frac{1}{4^{\frac{2}{5}}}???

## IMAGES

## VIDEO

## COMMENTS

Steps for Solving Equations with Fractional Exponents: 1. isolate the variable that has a fractional exponent. 2. convert from a fractional exponent to a

Learn how to deal with Rational Powers or Exponents. Exponents are shorthand for repeated multiplication of the same thing by itself.

Learn how to solve equations that involve fractional exponents. Also learn about base and exponent values. Easy Step-by-Step Explanation by

solve equations with fractional exponents. ... Fractional exponents with numerators other than 1 | Algebra I | Khan Academy. Khan Academy.

How would you simplify the following: (x^3)^(2/3) My first thought would be to multiply the exponents: 3/1 * 2/3 which would leave me with an exponent of 2.

When we divide fractional exponents with the same powers but different bases, we express it as a1/m ÷ b1/m = (a÷b)1/m.

The exponent of a number says how many times to use the number in a multiplication. So what does a fractional exponent mean?

Equations in which a variable expression is raised to a rational exponent can be solved by raising both sides of the equation to the reciprocal

For example, 3 x 3 x 3 x 3 can be written in exponential form as 34 where 3 is the base and 4 is the exponent. They are widely used in algebraic problems, and

Write the expression without fractional exponents. ... We can rewrite the expression by breaking up the exponent. ... We need to rationalize the