Simplifying Square Roots
The first step to solving square roots is knowing how to simplify them. For example, if you are given the square root √4, you can think of it as “the number that, when squared, equal four.” The correct answer would be 2, because when 2 is squared, it equals 2 X 2 = 4.
But what if the number under the square root sign isn’t a perfect square? In that case, you’ll need to factor it. So if you are given the problem √12, you would factor it to get √(2 X 2 X 3), or √(4 X 3). Then just take the √4 out, and write “2” instead, leaving only the “3” under the square root sign. That would leave you with 2√3.
Adding and Subtracting Square Roots
Believe it or not, adding and subtracting square roots or other irrational numbers is easy. Just treat the square root as a variable, such as "x" or "y." For example, if you are adding together 2√2 and 3√2, pretend that you are adding 2x and 3x:
2√2 + 3√2 = 5√2
Do the same thing for subtraction:
3√2 – 2√2 = 1√2 = √2
The next step is learning how to multiply square roots. To multiply square roots, make sure to separate the numbers outside the square root sign from those that are inside the square root sign. For example, to solve the problem 2√2 X 3√8, you would multiply the 2 and 3 together first, to get 6, and then you would multiply together the numbers inside of the square root and simplify your answer. So the problem would look like this:
2√2 X 3√8 = (2X3)√(2X8) = 6√16 = 6X4 = 24
Dividing By Square Roots
Dividing by square roots gets a bit more complicated. Sometimes, you can just cancel out the denominator or simplify it. For example, if you were given the problem √8/√2, you could divide the numerator and the denominator by √2, which would leave you with √4/1, or 2. So the problem would look like this:
√8/√2 = √(8/2)/√(2/2) = √4/1 = √4 = 2
You could also come across a more complicated difference, such as √2/√3. How can you simplify that? Remember one simple rule: the denominator can never be a radical (a square root). In order to get the square root out of the denominator, multiply both the numerator and the denominator by that square root. For example, in the problem √2/√3, you would multiply both the top and the bottom by √3. The result would look like this:
√2/√3 = √2/√3 X √3/√3 = √(2X3)/√(3X3) = (√6)/3
And that's your final answer.
Not sure if you've understood? Try some of these sample square root problems:
- √16 = ?
- √27 = ?
- 2√24 = ?
- 2√2 + 3√2 – 4√2 = ?
- 4√2 X √2 = ?
- √2 X 3√15 X √3 = ?
- √27/√3 = ?
- 2√3/√2 = ?
- 3√2/√3 = ?
- (5√3)/(3√5) = ?
Here are some additional resources you can use to learn more about square roots:
Purple Math. "Square Roots." https://www.purplemath.com/modules/radicals.htm
Sparknotes. "Exponents." https://www.sparknotes.com/math/algebra1/exponents/section1.html
Homeschool Math. "How to Calculate a Square Root Without a Calculator." https://www.homeschoolmath.net/teaching/square-root-algorithm.php
This post is part of the series: Math Study Guides
- Adding Irrational Numbers: A Step-by-Step Guide
- Two Techniques for Adding Mixed Fractions – With Examples
- Steps for Dividing Mixed Fractions with Examples and Resources
- Learn How to Solve Square Root Math Problems: Examples and Resources