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'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!!

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.

Is there an answer to x^3 without "x" being part of the answer?

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".

Main content

8th grade

Course: 8th grade > Unit 1

Lesson 6: Exponent properties intro

Exponent properties review

Google Classroom

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.

x, start superscript, n, end superscript, dot, x, start superscript, m, end superscript, equals, x, start superscript, n, plus, m, end superscript

Example

5, squared, dot, 5, start superscript, 5, end superscript, equals, 5, start superscript, 2, plus, 5, end superscript, equals, 5, start superscript, 7, end superscript

[Show me why this works.]

Practice

Problem 1.1

Current

Simplify.
Rewrite the expression in the form 8, start superscript, n, end superscript.

8, start superscript, 6, end superscript, dot, 8, start superscript, 4, end superscript, equals

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.

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

Example

start fraction, 3, start superscript, 8, end superscript, divided by, 3, squared, end fraction, equals, 3, start superscript, 8, minus, 2, end superscript, equals, 3, start superscript, 6, end superscript

[Show me why this works.]

Practice

Problem 2.1

Current

Simplify.
Rewrite the expression in the form 7, start superscript, n, end superscript.

start fraction, 7, start superscript, 7, end superscript, divided by, 7, cubed, end fraction, equals

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 parenthesis, x, start superscript, n, end superscript, right parenthesis, start superscript, m, end superscript, equals, x, start superscript, n, dot, m, end superscript

Example

left parenthesis, 8, squared, right parenthesis, cubed, equals, 8, start superscript, 2, dot, 3, end superscript, equals, 8, start superscript, 6, end superscript

[Show me why this works.]

Practice

Problem 3.1

Current

Simplify.
Rewrite the expression in the form 2, start superscript, n, end superscript.

left parenthesis, 2, start superscript, 4, end superscript, right parenthesis, squared, equals

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 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

Example

left parenthesis, 3, dot, 5, right parenthesis, start superscript, 6, end superscript, equals, 3, start superscript, 6, end superscript, dot, 5, start superscript, 6, end superscript

[Show me why this works.]

Practice

Problem 4.1

Current

Select the equivalent expression.

left parenthesis, 4, dot, 7, right parenthesis, start superscript, 8, end superscript, equals, question mark

(Choice A)
4, start superscript, 8, end superscript, dot, 7, start superscript, 8, end superscript
(Choice B)
4, start superscript, 4, end superscript, dot, 7, start superscript, 4, end superscript
(Choice C)
4, start superscript, 15, end superscript
(Choice D)
4, dot, 7, start superscript, 8, end superscript

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 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

Example

left parenthesis, start fraction, 7, divided by, 2, end fraction, right parenthesis, start superscript, 8, end superscript, equals, start fraction, 7, start superscript, 8, end superscript, divided by, 2, start superscript, 8, end superscript, end fraction

[Show me why this works.]

Practice

Problem 5.1

Current

Select the equivalent expression.

left parenthesis, start fraction, 6, divided by, 5, end fraction, right parenthesis, start superscript, 9, end superscript, equals, question mark

(Choice A)
start fraction, 6, start superscript, 9, end superscript, divided by, 5, start superscript, 9, end superscript, end fraction
(Choice B)
left parenthesis, 6, minus, 5, right parenthesis, start superscript, 9, end superscript
(Choice C)
6, start superscript, 4, end superscript
(Choice D)
start fraction, 6, start superscript, 9, end superscript, divided by, 5, end fraction

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

Want to join the conversation?

Sort by:

janana
Posted 7 years ago. Direct link to janana's post “I'm confused by the fact ...”
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?
Button navigates to signup pageComment on janana's post “I'm confused by the fact ...”
(27 votes)
Answer
- Hosannah H
  Posted 7 years ago. Direct link to Hosannah H's post “When you look at it, not ...”
  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!!
  Comment on Hosannah H's post “When you look at it, not ...”
  (147 votes)
Gabriel Zubovsky
Posted 5 years ago. Direct link to Gabriel Zubovsky's post “What's 0 to the 0th power...”
What's 0 to the 0th power ?
Button navigates to signup pageComment on Gabriel Zubovsky's post “What's 0 to the 0th power...”
(28 votes)
Answer
- Lovish Garg
  Posted 2 years ago. Direct link to Lovish Garg's post “Well it will be undefined...”
  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.
  Comment on Lovish Garg's post “Well it will be undefined...”
  (26 votes)
25jevel289
Posted 3 years ago. Direct link to 25jevel289's post “who added letters in math...”
who added letters in math i would like to have a conversation with them please
Button navigates to signup pageComment on 25jevel289's post “who added letters in math...”
(36 votes)
Answer
- Mikeala
  Posted 7 months ago. Direct link to Mikeala's post “François Viète is the per...”
  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.
  Button navigates to signup page
  (13 votes)
Victoria563
Posted 3 years ago. Direct link to Victoria563's post “Can an exponent have an e...”
Can an exponent have an exponent?
Button navigates to signup pageComment on Victoria563's post “Can an exponent have an e...”
(21 votes)
Answer
- THE BETTER WINCHESTER
  Posted 2 years ago. Direct link to THE BETTER WINCHESTER's post “Of course! It's mostly se...”
  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.
  Comment on THE BETTER WINCHESTER's post “Of course! It's mostly se...”
  (22 votes)
ɱ ä ɾ ɨ
Posted 4 months ago. Direct link to ɱ ä ɾ ɨ's post “If it's not: ```You'll ne...”
If it's not:
You'll never know when this will come up in life
It's:
You'll never know when you'll need this in life
Button navigates to signup pageComment on ɱ ä ɾ ɨ's post “If it's not: ```You'll ne...”
(33 votes)
Answer
izayah
Posted a year ago. Direct link to izayah's post “i wish we can go back to ...”
i wish we can go back to 1+1 and 2+2
Button navigates to signup pageComment on izayah's post “i wish we can go back to ...”
(23 votes)
Answer
- RAIDA YESSIN
  Posted 7 months ago. Direct link to RAIDA YESSIN's post “If you still using them d...”
  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
  Button navigates to signup page
  (4 votes)
Meng Fan
Posted 4 months ago. Direct link to Meng Fan's post “I am using Khan Academy t...”
I am using Khan Academy the first time. Khan Academy teaches very Amazingly.
Button navigates to signup pageComment on Meng Fan's post “I am using Khan Academy t...”
(21 votes)
Answer
- Manny
  Posted 2 months ago. Direct link to Manny's post “for real im doing this in...”
  for real im doing this in spare time
  Button navigates to signup page
  (1 vote)
kabeeralimuhammad11
Posted 5 months ago. Direct link to kabeeralimuhammad11's post “I am using Khan Academy f...”
I am using Khan Academy first time. Khan Academy teaches very Amazingly.
Button navigates to signup pageButton navigates to signup page
(22 votes)
Answer
hyost
Posted 3 months ago. Direct link to hyost's post “this is way to easy! I ha...”
this is way to easy! I have alr learned this and my mamma is making me do this :(
Button navigates to signup pageComment on hyost's post “this is way to easy! I ha...”
(16 votes)
Answer
dsnider
Posted 7 years ago. Direct link to dsnider's post “Is there an answer to x^3...”
Is there an answer to x^3 without "x" being part of the answer?
Button navigates to signup pageButton navigates to signup page
(6 votes)
Answer
- Kim Seidel
  Posted 7 years ago. Direct link to Kim Seidel's post “The only way you can get ...”
  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".
  Comment on Kim Seidel's post “The only way you can get ...”
  (13 votes)