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Square Root: Definition and Example | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Square RootSquare Root: Definition and ExampleTable of ContentsSquare Root Definition of Square Root

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 999 is 333 because 3×3=93 \times 3 = 93×3=9. We use the symbol  \sqrt{\ } ​ to show the square root of a number. When we write 9=3\sqrt{9} = 39​=3, we are saying that 333 is the square root of 999.

Every positive number has two square roots - one positive and one negative. For instance, both 333 and −3-3−3 are square roots of 999 because both 3×3=93 \times 3 = 93×3=9 and (−3)×(−3)=9(-3) \times (-3) = 9(−3)×(−3)=9. However, when we talk about "the square root" of a number, we usually mean the positive square root, which is called the principal square root. The square root of 000 is 000, and negative numbers don't have real square roots.

Examples of Square Root Example 1: Finding the Square Root of a Perfect Square Problem:

Find the square root of 646464.

Step-by-step solution:

Step 1, Break down 646464 into its prime factors.

64=2×2×2×2×2×2=2664 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^{6}64=2×2×2×2×2×2=26

Step 2, Group the prime factors in pairs.

64=(22)3=4364 = (2^{2})^{3} = 4^{3}64=(22)3=43

Step 3, Take out one factor from each pair.

64=26=23=8\sqrt{64} = \sqrt{2^{6}} = 2^{3} = 864​=26​=23=8

Step 4, Check your answer by multiplying it by itself.

8×8=648 \times 8 = 648×8=64, so 64=8\sqrt{64} = 864​=8.

Step 5, Therefore, the square root of 646464 is 888.

Example 2: Finding the Square Root of a Non-Perfect Square Problem:

Find the approximate value of 20\sqrt{20}20​.

Step-by-step solution:

Step 1, Look for the perfect squares closest to 202020.

16202516 162025

16=4\sqrt{16} = 416​=4 and 25=5\sqrt{25} = 525​=5

Step 2, Since 202020 is between 161616 and 252525, 20\sqrt{20}20​ must be between 444 and 555.

Step 3, We can get a better estimate by noticing that 202020 is closer to 161616 than to 252525.

20−16=420 - 16 = 420−16=4 and 25−20=525 - 20 = 525−20=5

Step 4, Since 202020 is a little bit more than 161616, 20\sqrt{20}20​ is a little bit more than 444. We can estimate 20≈4.5\sqrt{20} \approx 4.520​≈4.5

Step 5, For a more accurate value, we can use a calculator to find 20≈4.47\sqrt{20} \approx 4.4720​≈4.47.

Example 3: Simplifying Square Roots Problem:

Simplify 75\sqrt{75}75​.

Step-by-step solution:

Step 1, Break down 757575 into its prime factors.

75=3×25=3×5275 = 3 \times 25 = 3 \times 5^{2}75=3×25=3×52

Step 2, Separate factors into perfect squares and other factors.

75=3×52=3×52\sqrt{75} = \sqrt{3 \times 5^{2}} = \sqrt{3} \times \sqrt{5^{2}}75​=3×52​=3​×52​

Step 3, Simplify the square root of the perfect square.

52=5\sqrt{5^{2}} = 552​=5

Step 4, Write the final simplified form.

75=53\sqrt{75} = 5\sqrt{3}75​=53​

Step 5, Check your answer by squaring the simplified form.

(53)2=52×(3)2=25×3=75(5\sqrt{3})^{2} = 5^{2} \times (\sqrt{3})^{2} = 25 \times 3 = 75(53​)2=52×(3​)2=25×3=75

Comments(2)SShepherdLeoNovember 4, 2025I've used this square root def to help my students. It's clear & easy to understand. Great for making the concept click!

NNatureLover25September 16, 2025I loved how clear the square root definition was! I used the examples to help my son with his homework, and it really clicked for him. Practical tips like area calculations made it so relatable!

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