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### Course: MAP Recommended Practice>Unit 19

Lesson 2: Exponent properties intro

# Exponent properties review

Review the common properties of exponents that allow us to rewrite powers in different ways. For example, x²⋅x³ can be written as x⁵.
PropertyExample
${x}^{n}\cdot {x}^{m}={x}^{n+m}$${2}^{3}\cdot {2}^{5}={2}^{8}$
$\frac{{x}^{n}}{{x}^{m}}={x}^{n-m}$$\frac{{3}^{8}}{{3}^{2}}={3}^{6}$
${\left({x}^{n}\right)}^{m}={x}^{n\cdot m}$${\left({5}^{4}\right)}^{3}={5}^{12}$
$\left(x\cdot y{\right)}^{n}={x}^{n}\cdot {y}^{n}$$\left(3\cdot 5{\right)}^{7}={3}^{7}\cdot {5}^{7}$
${\left(\frac{x}{y}\right)}^{n}=\frac{{x}^{n}}{{y}^{n}}$${\left(\frac{2}{3}\right)}^{5}=\frac{{2}^{5}}{{3}^{5}}$

## Product of powers

This property states that when multiplying two powers with the same base, we add the exponents.
${x}^{n}\cdot {x}^{m}={x}^{n+m}$

### Example

${5}^{2}\cdot {5}^{5}={5}^{2+5}={5}^{7}$

### Practice

Problem 1.1
Simplify.
Rewrite the expression in the form ${8}^{n}$.
${8}^{6}\cdot {8}^{4}=$

Want to try more problems like these? Check out this exercise.

## Quotient of powers

This property states that when dividing two powers with the same base, we subtract the exponents.
$\frac{{x}^{n}}{{x}^{m}}={x}^{n-m}$

### Example

$\frac{{3}^{8}}{{3}^{2}}={3}^{8-2}={3}^{6}$

### Practice

Problem 2.1
Simplify.
Rewrite the expression in the form ${7}^{n}$.
$\frac{{7}^{7}}{{7}^{3}}=$

Want to try more problems like these? Check out this exercise.

## Power of a power property

This property states that to find a power of a power we multiply the exponents.
${\left({x}^{n}\right)}^{m}={x}^{n\cdot m}$

### Example

${\left({8}^{2}\right)}^{3}={8}^{2\cdot 3}={8}^{6}$

### Practice

Problem 3.1
Simplify.
Rewrite the expression in the form ${2}^{n}$.
${\left({2}^{4}\right)}^{2}=$

Want to try more problems like these? Check out this exercise.

## Power of a product

This property states that when taking the power of a product, we multiply the powers of the factors.
$\left(x\cdot y{\right)}^{n}={x}^{n}\cdot {y}^{n}$

### Example

$\left(3\cdot 5{\right)}^{6}={3}^{6}\cdot {5}^{6}$

### Practice

Problem 4.1
Select the equivalent expression.
$\left(4\cdot 7{\right)}^{8}=?$

Want to try more problems like these? Check out this exercise.

## Power of a quotient

This property states that when taking the power of a quotient, we divide the powers of the numerator and of the denominator.
${\left(\frac{x}{y}\right)}^{n}=\frac{{x}^{n}}{{y}^{n}}$

### Example

${\left(\frac{7}{2}\right)}^{8}=\frac{{7}^{8}}{{2}^{8}}$

### Practice

Problem 5.1
Select the equivalent expression.
${\left(\frac{6}{5}\right)}^{9}=?$

Want to try more problems like these? Check out this exercise.

## Want to join the conversation?

• I'm confused by the fact that all exponents to the 0th power equals 1. Why is this? Shouldn't the answer be zero instead?
• When you look at it, not really. Let's pick a small number: 2
2 to the power of 4=16
2 to the power of 3=8
2 to the power of 2=4
2 to the power of 1=2
Now when you look at these numbers, you should notice a pattern. 8/2=4, and 4/2=2. Now 2 divided by 2 would give us the answer to 2 to the power of 0, which is equal to 1.

Hope that you understand now. Good luck!!
• who added letters in math i would like to have a conversation with them please
• François Viète is the person you should speak to about adding numbers to math.
At the end of the 16th century, François Viète introduced the idea of representing known and unknown numbers by letters, nowadays called variables, and the idea of computing with them as if they were numbers—in order to obtain the result by a simple replacement.
• What's 0 to the 0th power ?
• Well it will be undefined. See this case
2^3=2*2*2 =8
2^2=2*2 =4
2^1=2 =2
2^0=1
The reason we get 2^0 is because for every 2^{n-1}, we are dividing the 2^n by 2, for example to get value of 2^0, we are dividing the 2^1=2 by the 2. The result is therefor 1.

But in case of 0, we will be dividing the 0 by the 0. Because 0^1=0 and then we will be diving by our base (which is 0), the result will be 0/0, which is undefined.
I hope you got my point.
Have a nice day.
• i wish we can go back to 1+1 and 2+2
• If you still using them don’t worry but to survive in this digital era and going to Mars you will need those more complicated mathematical expressions and equations to solve your everyday problems and you may solve one of the world problems one day you never know. My humble advice to anyone who is taking mathematics is to start from beginning and build up at your own pace so you can perceive it better and have as many examples as possible also research what you can't understand and try to solve as many problems as you can
• Can an exponent have an exponent?
• Of course! It's mostly seen in this form, though: (4^2)^3 where there is one exponent inside the parenthesis then outside the parenthesis there's another exponent, which applies to all parts inside the parenthesis, including the exponent inside.
• I am using Khan Academy first time. Khan Academy teaches very Amazingly.
• this is way to easy! I have alr learned this and my mamma is making me do this :(
• L couldn't be me
• Why do we have to learn all of this??