Square Root
In short: Square root of a number means a value that returns the number after being multiplied by itself, and the radical sign picks out the non-negative one, the principal root.
8th GradeNumbers and Operations8.EE.A.2
What is Square Root?
A square root reverses squaring. Since \(14 \times 14 = 196\), the root of 196 is 14, written \(\sqrt{196} = 14\). The radical sign demands the non-negative answer, known as the principal root.
Each positive number owns two roots, one positive and one negative, because \((-14)^2\) also equals 196. When an equation reads \(x^2 = 196\), both \(x = 14\) and \(x = -14\) work. Roots of numbers that are not squares of integers, like \(\sqrt{50}\), are irrational and get estimated.
How to spot it
- An area of a square is known and the side length is needed, such as a 196 square foot garden.
- An equation has the form x squared equals a number and the value of x must be found.
- A radical sign appears, possibly around a fraction or a decimal.
- A root such as the square root of 50 must be placed between two integers.
The method
- Ask which positive number multiplied by itself gives the radicand; test perfect squares like 144, 169, 196.
- If none fits, find the two perfect squares around the radicand and bracket the root between their roots.
- Refine by squaring a decimal guess, such as \(7.1^2 = 50.41\), until close enough.
- For \(x^2 = k\), remember both \(\sqrt{k}\) and \(-\sqrt{k}\) are solutions.
Square Root: solved example
Problem
A community plot is laid out as a square covering 196 square feet. How long is each edge? Then estimate the edge of a square plot covering 50 square feet.
Solution
The side \(s\) satisfies \(s^2 = 196\), so \(s = \sqrt{196} = 14\) feet.
For 50: \(7^2 = 49\) and \(8^2 = 64\) trap the edge between 7 and 8, and \(7.1^2 = 50.41\) shows it is just under 7.1. Side lengths: 14 feet and about 7.07 feet.
Try it
Question 1
Evaluate \(\sqrt{\frac{81}{121}}\).
Show the answer
\(\frac{\sqrt{81}}{\sqrt{121}} = \frac{9}{11}\).
Question 2
Estimate \(\sqrt{90}\) to the nearest tenth.
Show the answer
\(9^2 = 81\) and \(10^2 = 100\); \(9.5^2 = 90.25\), slightly over, so \(\sqrt{90} \approx 9.5\).
Square Root vs. Perfect Square
A perfect square is a number owning a whole-number root, like 196. The square root is the operation, or its output, that travels the opposite way. The root of a perfect square is a whole number; the root of anything else is a decimal that never ends or repeats.
Keep going
Lessons
Worksheets and quizzes
- Square Roots and Cube Roots – Worksheet
- Square Roots and Cube Roots – Quiz
- Rational and Irrational Numbers – Worksheet
- Rational and Irrational Numbers – Quiz
Related words
Whole dictionary: A–Z · 8th Grade · Numbers and Operations