Pre-Algebra:
Square and Cube Roots

1.  
\(\sqrt{{x}^{2}}\)
  Solution
2.  
\(\sqrt{162}\)
  Solution
3.  
\(\sqrt{720}\)
  Solution
4.  
\(\frac{\sqrt{20}}{2}\)
  Solution
5.  
\(\frac{\sqrt{1.44}}{2}\)
  Solution
6.  
\(\sqrt{{(2\times 3)}^{2}-4\times {3}^{2}}\)
  Solution
7.  
\({25}^{0.5}\)
  Solution
8.  
\(\sqrt[3]{64}\)
  Solution
9.  
\(\sqrt[3]{125}\)
  Solution
10.  
\(\sqrt[3]{4}\sqrt[3]{4}\sqrt[3]{4}\)
  Solution

Square and Cube Roots - Introduction

Square and cube roots — also known as radicals — are powerful concepts in mathematics. It is crucial in the concept of standard deviation in the field of probability theory and statistics. It is also critical in the formula for roots of a quadratic equation.
Taking the square or cube root is the opposite of squaring or cubing, which occurs when numbers are multiplied by themselves (squared), or multiplied by themselves twice (cubed).

How to Find Roots

For example, if you see the square root
\(\sqrt{100}\)
, you are looking for the number that yields
\(100\)
when multiplied by itself. The answer is
\(10\)
, because
\(10\times 10=100\)
. This is called a perfect square because
\(10\)
is a whole number rather than a decimal. The same applies to cube roots: The root of
\(\sqrt[3]{27}\)
is
\(3\)
because
\(3\times 3\times 3=27\)
. Similarly, this is a perfect cube, because the answer is a whole number.
Yet, what happens if you are dealing with a square or cube root that does not yield a whole number? Let’s look at how we can simplify the root in these cases.

How to Simplify Roots

Let's use a method that involves prime factors.
Example: Simplify the square root
\(\sqrt{162}\)
.
Solution: Start by rewriting the radical as its prime factors, which are
\(\sqrt{2\times 3\times 3\times 3\times 3}\)
. Then, group the same prime factors into pairs, then rewrite them as squares to give
\(\sqrt{{3}^{2}\times {3}^{2}\times 2}\)
. Next, use
\(\sqrt{{x}^{2}}=x\)
to simplify the root and give
\((3\times 3)\sqrt{2}\)
. Finally, simplify again to produce
\(9\sqrt{2}\)
.
For more detail, see the full solution here.

What's Next

We hope you now understand the basic concept of roots, as well as the basic techniques to find roots and to simplify them. You can also try other practice problems to hone your skills and on other math topics. Have a difficult homework problem about square and cute roots? Try Cymath’s square and cube root calculator - you can simply enter your number and the app will provide detailed solution steps.
At Cymath, we have designed our app to help students succeed with clear, easy-to-understand solutions, so that they can get the explanations and skills they need to improve overall math ability and confidence. You can also get more help with Cymath Plus.
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