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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, start superscript, n, end superscript, dot, x, start superscript, m, end superscript, equals, x, start superscript, n, plus, m, end superscript2, 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 superscriptstart 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 superscriptleft 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 superscriptleft 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 fractionleft 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

Practice

Problem 1.1
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

Practice

Problem 2.1
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

Practice

Problem 3.1
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

Practice

Problem 4.1
Select the equivalent expression.
left parenthesis, 4, dot, 7, right parenthesis, start superscript, 8, end superscript, equals, question mark
Choose 1 answer:
Choose 1 answer:

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

Practice

Problem 5.1
Select the equivalent expression.
left parenthesis, start fraction, 6, divided by, 5, end fraction, right parenthesis, start superscript, 9, end superscript, equals, question mark
Choose 1 answer:
Choose 1 answer:

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

Want to join the conversation?

  • blobby green style avatar for user Victoria563
    Can an exponent have an exponent?
    (20 votes)
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  • piceratops tree style avatar for user Gabriel Zubovsky
    What's 0 to the 0th power ?
    (2 votes)
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    • orange juice squid orange style avatar for user Lovish Garg
      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)
  • blobby blue style avatar for user Montilla, Andrea
    can an exponent have an exponent
    (6 votes)
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    • mr pants purple style avatar for user 50024
      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)
  • blobby green style avatar for user daledrick jackson
    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)
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  • blobby green style avatar for user izayah
    i wish we can go back to 1+1 and 2+2
    (7 votes)
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  • blobby green style avatar for user Tashi Tsering
    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)
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  • blobby green style avatar for user Ana Morales
    How would you solve: x^-7/8y^2 (times ) 8/ x^8y^-2?
    (6 votes)
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  • purple pi teal style avatar for user dsnider
    Is there an answer to x^3 without "x" being part of the answer?
    (0 votes)
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  • blobby green style avatar for user csb2020
    can you explain how to resolve a^8b^-9c^2/a^-3b^6c^4
    thank you
    (0 votes)
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    • piceratops ultimate style avatar for user Hecretary Bird
      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)
  • purple pi purple style avatar for user louisaandgreta
    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)
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