Study Guide

Basic Operations - Roots

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Roots are sort of like the opposites of powers

The square root of 9, written as 9^1/2, means "what number times itself gives an answer of 9?"  

  • 32, or 3 to the second power (3 × 3) is 9.
  • So the square root of 9 is 3.

The cube root of 8, written as 8^1/3, means "what number times itself three times is 8?" 

  • 23, or 2 to the third power (2 × 2 × 2) is 8.
  • So the cube root of 8 is 2.

The order of operation (PEMDAS) always applies, even to roots. When there's an operation to simplify underneath the root symbol, it may or may not be in parentheses, but we need to simplify the operation first as if it were in parentheses, then take the root. Here's an example.

To solve this guy, we add the stuff inside the square root symbol first.


Now we grab the square root. What number multiplied by itself gives us 25? Hmm...we know 5 × 5 = 25, so the square root of 25 must be 5.



Here's an important tidbit that we can't stress enough: You can't take the square root of a negative number.

A number times itself can never be negative because a negative times a negative is positive, and a positive times a positive is also positive. There's no real number in the entire universe that we can square to get a negative answer. So there's no answer to the problem (-49)^1/2 , since 7 × 7 = +49 and (-7) × (-7) = +49 too.

However, you can take the cube root of a negative number because if we multiply a negative number by itself three times, we still get a negative answer. Three negatives make a negative.

For example, since (-3) × (-3) × (-3) = -27, the cube root of -27 is -3. In math-ese:

-27^1/3 = -3

A Mathematical "Undo" Button

Remember how at the beginning of this very engaging reading we said that roots are sort of like the opposites of powers? Now that we know what roots are, we can illustrate.

What do we get when we square the square root of 25?


First we grab the square root of 25 to get 5, and then we square that 5, which gives us 25 again.

()2 = (5)2 = 25

These are inverse operations because whatever the first operation (square root) does to our 25, the second operation (squared) undoes it. This also works the other way: if we took the square root of 52, we'd get 5 back again. It feels a bit like a dog chasing its tail, but that's inverse operations for ya.

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