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Exponents & radicals: FAQ

Frequently asked questions about exponents & radicals

What are exponent properties?

We can use these properties of exponents to help us simplify expressions involving exponents:
Product rule: xa×xb=xa+b. For example, x2×x3=x5.
Power rule: (xa)b=xab. For example, (x2)3=x6.
Quotient rule: xaxb=xab. For example, x5x2=x3.
Zero exponent rule: x0=1. For example, 70=1.

What are radicals?

A radical is a symbol that we use to write square roots, cube roots, and other roots. For example, 81 is the square root of 81, or the number we can multiply by itself to get 81. Another example is A83, which is the cube root of 8, or the number we can multiply by itself to get 8.
Practice with our Square roots exercise.
Practice with our Cube roots exercise.

What are some common ways to simplify square roots?

One common way to simplify square roots is to factor the radicand (the number inside the square root symbol) into perfect squares. For example, 50 can be simplified by factoring 50 into 25 times 2:
50=25×2=25×2=52

Want to join the conversation?

  • marcimus purple style avatar for user Evan Evan
    There's a mistake on this page where it says "the cubed root of 8 is the number we can multiply by itself to get 8". It should actually say the number we can multiply by itself 3 times to get 8 (because it's the cubed root, not the square root)
    (40 votes)
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  • aqualine ultimate style avatar for user Qyqjaari The GOAT
    help me I have a high school entrance exam soon, what should I study to get a good score?
    (6 votes)
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  • leafers ultimate style avatar for user Evangeline Viklund
    Is this correct?

    Product: if you multiply a number with an exponent by the same number but with a different exponent, you add the two exponents together, leaving the whole number as is.

    Power: If you multiply a number with an exponent by an exponent, you multiply the two exponents and leave the base number alone.

    Quotient: Since division is the opposite of multiplication, naturally the addition power of multiplying exponents becomes subtraction when you divide them. (Be careful with negative numbers! They have tripped me up several times.)

    Zero exponent rule: strange and tricky. imagine it like this: you have a number you want to multiply exponentially, say, 7. Instead of thinking this as the base, think of the neutral base as 1. If 1 is multiplied exponentially by 7 zero times, it wasn't multiplied at all. so it remains its original self: 1. If it was multiplied once, it would be 7. (This is just an idea of how to think about it! please someone comment if you see a flaw in my logic.)

    Your friendly neighborhood Spider-girl
    (2 votes)
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  • blobby green style avatar for user mwyatt27
    What's a rational number?
    (1 vote)
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    • hopper cool style avatar for user vorb
      @EllieValeska223

      Great Question! A rational number is any number that can be written as a fraction, where both the numerator (the top number) and the denominator (the bottom number) are integers, and the denominator is not equal to zero. In other words, a rational number can be expressed as p/q, where p and q are both integers and q ≠ 0.

      Hope this helps.
      (3 votes)
  • blobby green style avatar for user clairey
    Can I find out which perfect squares I use to fully simplify and replace the radicand? For example, the square root of 160 could be simplified like 2 and the square root of 40, but the fully simplified answer is 4 to the square root of 10.
    (1 vote)
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  • area 52 yellow style avatar for user _
    Is this a chat room?
    (1 vote)
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  • blobby green style avatar for user Slay
    hey.... how can i tell the different rules apart?
    (1 vote)
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  • blobby green style avatar for user LiliaH
    Why do we do this
    (0 votes)
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  • blobby green style avatar for user Natalie
    confused
    (0 votes)
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  • piceratops ultimate style avatar for user enxyte
    There is an issue with the wording in the penultimate paragraph of the text.

    The wording states for cube roots there is a number that when multiplied by itself twice equals 8 as in a * a = a^2 = 8 instead of correctly asserting it is three times as in a * a * a = a^3 = 8.

    In other words: the text implies it is a square root which would render the result equivalent to 2.8284... * 2.8284... = 2.8284...^2 = 8. The cube root of 8 is 2 since 2 * 2 * 2 = 2^3 = 8.
    (0 votes)
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