The Basics Of Square Roots (Examples & Answers)

Square roots are often found in math and science problems, and any student needs to pick up the basics of square roots to tackle these questions. Square roots ask "what number, when multiplied by itself, gives the following result," and as such working them out requires you to think about numbers in a slightly different way. However, you can easily understand the rules of square roots and answer any questions involving them, whether they require direct calculation or just simplification.

Square roots are the opposite of "squaring" a number, or multiplying it by itself. For example, three squared is nine (3² = 9), so the square root of nine is three. In symbols, this is

\(\sqrt{9} = 3\)

The "√" symbol tells you to take the square root of a number, and you can find this on most calculators.

Remember that every number actually has ​two​ square roots. Three multiplied by three equals nine, but negative three multiplied by negative three also equals nine, so

\(3^2 = (-3)^2 = 9 \text{ and } \sqrt{9} = ±3\)

with the ± standing in for "plus or minus." In many cases, you can ignore the negative square roots of numbers, but sometimes it's important to remember that every number has two roots.

You may be asked to take the "cube root" or "fourth root" of a number. The cube root is the number that, when multiplied by itself twice, equals the original number. The fourth root is the number that when multiplied by itself three times equals the original number. Like square roots, these are just the opposite of taking the power of numbers. So, 3³ = 27, and that means the cube root of 27 is 3, or

\(\sqrt[3]{27} = 3\)

The "∛" symbol represents the cube root of the number that comes after it. Roots are sometimes also expressed as fractional powers, so

\(\sqrt{x} = x^{1/2} \text{ and } \sqrt[3]{x} = x^{1/3}\)

One of the most challenging tasks you may have to perform with square roots is simplifying large square roots, but you just need to follow some simple rules to tackle these questions. You can factor square roots in the same way as you factor ordinary numbers. So for example 6 = 2 × 3, so

\(\sqrt{6} = \sqrt{2} × \sqrt{3}\)

Simplifying larger roots means taking the factorization step by step and remembering the definition of a square root. For example, √132 is a big root, and it might be hard to see what to do. However, you can easily see it's divisible by 2, so you can write

\(\sqrt{132} = \sqrt{2} \sqrt{66}\)

However, 66 is also divisible by 2, so you can write:

\(\sqrt{2} \sqrt{66} = \sqrt{2} \sqrt{2} \sqrt{33}\)

In this case, a square root of a number multiplied by another square root just gives the original number (because of the definition of square root), so

\(\sqrt{132} = \sqrt{2} \sqrt{2} \sqrt{33} = 2 \sqrt{33}\)

In short, you can simplify square roots using the following rules

\(\sqrt{a × b} = \sqrt{a} × \sqrt{b}\)
\(\sqrt{a} × \sqrt{a} = a\)

Using the definitions and rules above, you can find the square roots of most numbers. Here are some examples to consider.

This can't be found directly because it isn't the square root of a whole number. However, using the rules for simplification gives:

\(\sqrt{8} = \sqrt{2} \sqrt{4} = 2 \sqrt{2}\)

This makes use of the simple square root of 4, which is √4 = 2. The problem can be solved exactly using a calculator, and √8 = 2.8284....

Using the same approach, try to work out the square root of 12. Split the root into factors, and then see if you can split it into factors again. Attempt this as a practice problem, and then look at the solution below:

\(\sqrt{12} = \sqrt{2} \sqrt{6} = \sqrt{2} \sqrt{2} \sqrt{3} = 2 \sqrt{3}\)

Again, this simplified expression can either be used in problems as needed, or calculated exactly using a calculator. A calculator shows that

\(\sqrt{12} = 2\sqrt{3} = 3.4641....\)

The square root of 20 can be found in the same way:

\(\sqrt{20} = \sqrt{2} \sqrt{10} = \sqrt{2} \sqrt{2} \sqrt{5}=2 \sqrt{5} = 4.4721....\)

Finally, tackle the square root of 32 using the same approach:

\(\sqrt{32} = \sqrt{4} \sqrt{8}\)

Here, note that we already calculated the square root of 8 as 2√2, and that √4 = 2, so:

\(\sqrt{32} = 2×2 \sqrt{2} = 4 \sqrt{2} = 5.657....\)

Square Root of a Negative Number

Although the definition of a square root means that negative numbers shouldn't have a square root (because any number multiplied by itself gives a positive number as a result), mathematicians encountered them as part of problems in algebra and devised a solution. The "imaginary" number ​i​ is used to mean "the square root of minus 1" and any other negative roots are expressed as multiples of ​i​. So

\(\sqrt{-9} = \sqrt{9} × i = ±3i\)

These problems are more challenging, but you can learn to solve them based on the definition of ​i​ and the standard rules for roots.

Test your understanding of square roots by simplifying as needed and then calculating the following roots:

\(\sqrt{50}\)
\(\sqrt{36}\)
\(\sqrt{70}\)
\(\sqrt{24}\)
\(\sqrt{27}\)

Try to solve these before looking at the answers below:

\(\sqrt{50} = \sqrt{2} \sqrt{25} = 5 \sqrt{2} = 7.071\)
\(\sqrt{36} = 6\)
\(\sqrt{70} = \sqrt{7} \sqrt{10} = \sqrt{7} \sqrt{2} \sqrt{5} = 8.637\)
\(\sqrt{24} = \sqrt{2} \sqrt{12} = \sqrt{2} \sqrt{2} \sqrt{6} = 2 \sqrt{6} = 4.899\)
\(\sqrt{27} = \sqrt{3} \sqrt{9} = 3 \sqrt{3} = 5.196\)