Square Roots
A square root undoes squaring.
About this topic
A square root undoes squaring: √49 = 7 because 7² = 49. It asks "what number multiplied by itself gives this?" Every positive number has two square roots, one positive and one negative, but the √ symbol denotes the positive principal root.
Perfect squares such as 1, 4, 9, 16 and 25 have whole-number roots. Others, like √2, are irrational and can only be approximated. To estimate √20, notice it lies between √16 = 4 and √25 = 5, and closer to 4, so about 4.5. Square roots simplify when you pull out perfect square factors: √72 = √(36 × 2) = 6√2.
Geometrically, the square root of an area gives the side length of a square. A square garden of 81 square metres has sides of 9 metres. Square roots appear in the Pythagorean theorem, distance calculations and the quadratic formula, and in statistics through the standard deviation. In a right triangle with legs 3 and 4, the hypotenuse is √(9 + 16) = 5.
Common mistakes include thinking the root of a sum equals the sum of roots: √(9 + 16) is 5, not 3 + 4 = 7. Another is believing the square root of a negative number is a real number. For real numbers it does not exist, which led mathematicians to invent imaginary numbers later. Also remember that √x² equals the absolute value of x, not simply x.
Key ideas
- √49 = 7 because 7² = 49
- Perfect squares have whole roots
- Square roots of non-squares are irrational
Quick check
Use cases: where Square Roots helps
- Homework: review the key idea behind Square Roots before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Square Roots without notes.
- Visual learning: play with the interactive lab to see how Square Roots changes when you adjust its inputs.
- Teaching a friend: explain Square Roots in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Square Roots?
A square root undoes squaring.
What will I learn in Square Roots?
√49 = 7 because 7² = 49. Perfect squares have whole roots. Square roots of non-squares are irrational.
Is Square Roots part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Square Roots lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Square Roots?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.