We simplify expressions in math all the time, like when reducing fractions or combining like terms in algebra. We do it to make numbers easier to work with, easier to compare, and easier to use in further calculations. The same idea works for square roots.
Today, we’ll walk you through why we simplify square roots, how to do it step by step, and give you a chance to practice. We'll also answer the questions students ask most about this topic along the way.
A square root of a number is a value that, when multiplied by itself, gives that number. In other words, it answers one question: what number times itself equals this?
For example:
\(\sqrt{9} = 3\), because 3 × 3 = 9
\(\sqrt{25} = 5\), because 5 × 5 = 25
\(\sqrt{64} = 8\), because 8 × 8 = 64
These are called perfect squares or numbers whose square roots are whole numbers.
Not every number works out that cleanly, though. \(\sqrt{2}\), \(\sqrt{5}\), and \(\sqrt{72}\) don't give us whole numbers. And that's exactly where simplification comes in.
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When a square root doesn't give us a whole number, the result can be difficult to read, compare, or use in further calculations. Simplification fixes that.
Take \(\sqrt{72}\). On its own, that number is hard to work with. But once we simplify it to \(6\sqrt{2}\), we can immediately see that it's 6 times the square root of 2, which is a much cleaner expression that's easier to handle in equations and build on in more advanced math.
Simplifying also helps us spot relationships between numbers that wouldn't be obvious otherwise. Two square roots that look completely different might simplify to the same thing, and we'd never know without simplifying.
Let's use \(\sqrt{72}\) as our example, the same one from before.
What we need to do here is break the number inside the root into two factors: the largest perfect square that divides into it, and what's left over.
We look inside the root and ask, "Can we write 72 as a product of two numbers, where one is a perfect square?
72 = 36 × 2
36 is a perfect square and the largest one that divides evenly into 72, so that's our factor.
\(\sqrt{72} = \sqrt{(36 × 2)}\)
We separate the two factors, giving each its own root symbol:
\(\sqrt{(36 × 2)} = \sqrt{36} × \sqrt{2}\)
\(\sqrt{36} = {6}\), so we bring that out in front and leave \(\sqrt{2}\) as it is:
\(\sqrt{72} = 6\sqrt{2}\)
That's the simplified form, and from here, it's much easier to use in any calculation.
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We'll work through two more examples together: one to reinforce the steps, and one with a slightly bigger number to build confidence before you go solo.
What perfect square hides inside 50? Let's find out.
Can we write 50 as a product of two numbers, where one is a perfect square?
50 = 25 × 2
25 is a perfect square and the largest one that divides evenly into 50.
\(\sqrt{50} = \sqrt{(25 × 2)}\)
\(\sqrt{(25 × 2)} = \sqrt{25} × \sqrt{2}\)
\(\sqrt{25} = 5\), so 5 comes out in front:
\(\sqrt{50} = 5\sqrt{2}\)
Let's try a slightly bigger number.
Can we write 108 as a product of two numbers, where one is a perfect square?
108 = 36 × 3
36 is a perfect square and the largest one that divides evenly into 108.
\(\sqrt{108} = \sqrt{(36 × 3)}\)
\(\sqrt{(36 × 3)} = \sqrt{36} × \sqrt{3}\)
\(\sqrt{36} = 6\), so 6 comes out in front:
\(\sqrt{108} = 6\sqrt{3}\)
Now you take the wheel. Use what we've worked through today and simplify these square roots on your own.
Challenge 1: Simplify \(\sqrt{18}\)
Challenge 2: Simplify \(\sqrt{45}\)
Challenge 3: Simplify \(\sqrt{98}\)
Challenge 4: Simplify \(\sqrt{200}\)
Once you're done, scroll down to the bottom of the guide to check your answers.
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These are the questions we get from our students when we cover simplifying square roots.
Yes. For a variable under a root sign, we look for the largest even power, since even powers are perfect squares.
For example, \(\sqrt{x^4} = x^2\) because \(x^2 × x^2 = x^4\).
When variables and numbers appear together under the root, we simplify each part separately and combine the results. This typically comes up in algebra, once students are comfortable simplifying square roots with numbers alone.
It depends. If the square roots are like terms, meaning they have the same number under the root sign, they can be added or subtracted directly. \(\sqrt{5} + \sqrt{5} = 2\sqrt{5}\), for example.
But if the roots look different, it's always worth simplifying first. Two square roots that appear different might simplify to the same form, making addition or subtraction possible after all.
\(\sqrt{8} + \sqrt{2}\) looks like two different terms, but \(\sqrt{8}\) simplifies to \(2\sqrt{2}\). Then, \(2\sqrt{2} + \sqrt{2} = 3\sqrt{2}\).
So the short answer: simplify first, then check if the terms can be combined
Then the square root is already in its simplest form. Numbers like \(\sqrt{2}\), \(\sqrt{3}\), \(\sqrt{5}\), and \(\sqrt{7}\) have no perfect square factors other than 1, so there's nothing to pull out from under the root sign. We leave them exactly as they are, as they're already as simple as they can get.
Most students are introduced to simplifying square roots in 8th grade, when they begin working more formally with radicals and irrational numbers.
The concept builds on a solid understanding of perfect squares and factors, which students typically develop in 6th and 7th grade. From there, simplifying square roots becomes an important skill in algebra and geometry.
Mathnasium uses personalized learning plans and interactive teaching techniques to help students build a deep understanding of any math skill, including simplifying square roots.
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If you’ve given our challenges a try, check your answers here.
Challenge 1: \(\sqrt{18} = \sqrt{(9 × 2)} = 3\sqrt{2}\)
Challenge 2: \(\sqrt{45} = \sqrt{(9 × 5)} = 3\sqrt{5}\)
Challenge 3: \(\sqrt{98} = \sqrt{(49 × 2)} = 7\sqrt{2}\)
Challenge 4: \(\sqrt{200} = \sqrt{(100 × 2)} = 10\sqrt{2}\)
Answer a few questions to see how it works
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