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## 8th grade

### Unit 1: Lesson 6

Exponent properties intro# Exponent properties review

CCSS.Math:

Review the common properties of exponents that allow us to rewrite powers in different ways. For example, x²⋅x³ can be written as x⁵.

Property | Example |
---|---|

x, start superscript, n, end superscript, dot, x, start superscript, m, end superscript, equals, x, start superscript, n, plus, m, end superscript | 2, cubed, dot, 2, start superscript, 5, end superscript, equals, 2, start superscript, 8, end superscript |

start fraction, x, start superscript, n, end superscript, divided by, x, start superscript, m, end superscript, end fraction, equals, x, start superscript, n, minus, m, end superscript | start fraction, 3, start superscript, 8, end superscript, divided by, 3, squared, end fraction, equals, 3, start superscript, 6, end superscript |

left parenthesis, x, start superscript, n, end superscript, right parenthesis, start superscript, m, end superscript, equals, x, start superscript, n, dot, m, end superscript | left parenthesis, 5, start superscript, 4, end superscript, right parenthesis, cubed, equals, 5, start superscript, 12, end superscript |

left parenthesis, x, dot, y, right parenthesis, start superscript, n, end superscript, equals, x, start superscript, n, end superscript, dot, y, start superscript, n, end superscript | left parenthesis, 3, dot, 5, right parenthesis, start superscript, 7, end superscript, equals, 3, start superscript, 7, end superscript, dot, 5, start superscript, 7, end superscript |

left parenthesis, start fraction, x, divided by, y, end fraction, right parenthesis, start superscript, n, end superscript, equals, start fraction, x, start superscript, n, end superscript, divided by, y, start superscript, n, end superscript, end fraction | left parenthesis, start fraction, 2, divided by, 3, end fraction, right parenthesis, start superscript, 5, end superscript, equals, start fraction, 2, start superscript, 5, end superscript, divided by, 3, start superscript, 5, end superscript, end fraction |

*Want to learn more about these properties? Check out this video and this video.*

## Product of powers

This property states that when multiplying two powers with the same base, we add the exponents.

### Example

### Practice

*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.

### Example

### Practice

*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.

### Example

### Practice

*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.

### Example

### Practice

*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.

### Example

### Practice

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

## Want to join the conversation?

- Can an exponent have an exponent?(20 votes)
- 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.(16 votes)

- What's 0 to the 0th power ?(2 votes)
- 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.(15 votes)

- can an exponent have an exponent(6 votes)
- 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.(3 votes)

- 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?(4 votes)
- When you look at it, not really. Lets 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 4/2=2 now two divided by two would give us the answer to 2 to the power of 0 which is equal to 1.(8 votes)

- i wish we can go back to 1+1 and 2+2(7 votes)
- 1) a^2 x a^3 = a^2+3.

2) a^2 x b^2 = axb^2

I got the both rules, but i question is what if both the Base as well as exponent are same like ( a^3 x a^3 = ?) do we apply both rule or just one rule takes precedence and another rule wont apply.(4 votes)- The rules give you equivalent answers (which is a good thing!).

Applying your first rule:

a^3 x a^3 = a^(3+3) = a^6

Applying your second rule:

a^3 x a^3 = (a^3)^2 = a^(3 x 2) = a^6(6 votes)

- How would you solve: x^-7/8y^2 (times ) 8/ x^8y^-2?(6 votes)
- Is there an answer to x^3 without "x" being part of the answer?(0 votes)
- The only way you can get an answer for x^3 without have "x" be part of the answer is if you know the value of "x". For example, if someone says x=4, then we can find x^3.

x^3 = 4^3 = 64. Otherwise, you are stuck with the "x".(10 votes)

- can you explain how to resolve a^8b^-9c^2/a^-3b^6c^4

thank you(0 votes)- First, let's tackle all the negative exponents. Since people don't like them, we have to switch them to the other side of the division and then cancel out the negative. From:

a^8 * b^(-9) * c^2 / a^(-3) * b^6 * c^4

We get:

a^8 * c^2 ***a^3**/ b^6 * c^4 ***b^9**

Now, we know that x^a * x^b = x^(a+b), and that x^a / x^b = x^(a-b). We can apply this to each variable and combine the exponents.

a^(8+3) / b^(6+9) * c^2

Simplify this, and we have our final answer:

a^11 / (b^15 * c^2)(7 votes)

- I get confused because there seems to be an inconsistency in the technique. Sometimes you can distribute the exponent, specifically 2, like here but earlier when we were learning how to factorize quadratics and how to check that the factorization was correct you had to double what was in the parentheses otherwise you wouldn’t get back to the initial equation if you just distributed the exponent 2.

But is this below correct:

You can essentially distribute an exponent onto the elements in parentheses as long as the elements are multiplied or being divided.

But if the elements in parentheses are added or subtracted then you can’t.

For instance

(4x times 3)^2. Here you can distribute the 2 exponent onto the 4 and the x and the 3

Whereas in (4x+3)^2 you can’t distribute the square you have to write (4x+3)(4x+3)

Correct?(1 vote)- Yes, that is correct. Exponents are shorthand for repetititve multiplication. So, you can distribute them to factors (items bein multiplied). You can't apply them to terms (items being added/subtracted).(4 votes)