A square root of a number is a value that, multiplied by itself, gives that number. The square root of 36 is 6 because 6 × 6 = 36. Every positive number has two square roots, such as 6 and −6, but the radical sign √ means the principal square root, which is the non-negative one.
In Spanish
raíz cuadrada
Also called
radical, root, principal square root
Grade level
6-8
Grade 8Algebra 1CCSS 8.EE.A.2GEDSATACT
Square Root in a picture
The square root of an area is the side of the square: √25 = 5. For a number that is not a perfect square, find the two perfect squares around it: √20 is between 4 and 5.
Formal definition and notation
For a real numbera ≥ 0, the principal square root √a is the unique real number b ≥ 0 with b2 = a. The equation x2 = a has the two solutions x = ±√a when a > 0, one solution when a = 0, and no real solution when a < 0.
Symbol
Read it as
Meaning
√a
the square root of a
the principal (non-negative) square root of a; the sign is called a radical
for a, b ≥ 0; used to simplify: √72 = √36 × √2 = 6√2
√ab = √a√b
for a ≥ 0, b > 0: √925 = 35
√a + b ≠ √a + √b
there is no sum rule: √9 + 16 = 5, not 7
Square roots of perfect squares (1, 4, 9, 16, 25, …) are whole numbers. Square roots of other whole numbers, such as √2 ≈ 1.414, are irrational.
Try it: build a square
Interactive
Drag the corner (or use the arrow keys) until the length and the width are equal. The area is then a perfect square, and the side length is its square root: a 7 by 7 square has area 49, so the square root of 49 is 7.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Square Root examples, step by step
Example 1: A perfect square
Warm-up
Find √121.
Look for a number that, multiplied by itself, gives 121.
10 × 10 = 100 is too small; 11 × 11 = 121.
The symbol asks for the non-negative root, so the answer is 11, not −11.
Answer: √121 = 11.
Example 2: Estimate a square root
Core
Estimate √50 to the nearest tenth without a calculator.
Find the perfect squares on either side: 49 = 72 and 64 = 82, so √50 is between 7 and 8.
50 is very close to 49, so the root is just above 7.
Test: 7.12 = 50.41, slightly more than 50, and 7.02 = 49, less than 50.
50 is much closer to 50.41 than to 49, so round to 7.1. (A calculator gives 7.071…)
Answer: √50 is about 7.1.
Example 3: Side length from area
Test-style
A square garden has an area of 196 square feet. How many feet of fencing are needed to go all the way around it?
The side of a square is the square root of its area: s = √196.
14 × 14 = 196, so s = 14 feet.
Fencing goes around all 4 sides: 4 × 14 = 56 feet.
Trap to avoid: 196 is the area in square feet, not the fence length.
Answer: The fence must be 56 feet long (each side is 14 feet).
Common mistakes
✗ Mistake√49 = ± 7, or √49 = −7.
✓ Instead√49 = 7. The principal square root is non-negative. The equation x2 = 49 has two solutions, ± 7.
The radical sign names one number, the non-negative root. Write −√49 or ±√49 when you want the negative one too.
✗ Mistake√16 = 8, because 16 divided by 2 is 8.
✓ Instead√16 = 4, because 4 × 4 = 16.
A square root asks which number times itself gives 16. Halving is a different operation; 8 times 8 is 64.
✗ Mistake√9 + 16 is √9 + √16, which is 7.
✓ InsteadAdd first: √9 + 16 = √25 = 5.
Square roots split over multiplication, not over addition. The radical works like parentheses: do what is inside first.
✗ Mistake√−25 = −5.
✓ Instead√−25 is not a real number. Note (−5)2 = 25, not −25.
Any real number times itself is positive or zero, so no real number squares to a negative.
Not to be confused with
Square Rootvs.Squaring a number
Squaring multiplies a number by itself: 62 = 36. The square root goes the other way: √36 = 6.
A cube root is a number used three times as a factor: the cube root of 27 is 3 because 3 × 3 × 3 = 27. Cube roots of negative numbers exist; square roots do not.
An exponent tells how many times to multiply a base. The square root is the inverse of the exponent 2, and can be written as the exponent 12: 912 = 3.
Quick check: 3 questions
Question 1 of 3
What is √49?
Show answer and explanation
Answer: A.7 × 7 = 49, and the radical sign means the non-negative root, so √49 = 7. Choices B and C include the negative root, which belongs to the equation x2 = 49, and D divides by 2.
Question 2 of 3
Between which two consecutive whole numbers is √90?
Show answer and explanation
Answer: B.92 = 81 and 102 = 100, and 90 is between 81 and 100, so √90 is between 9 and 10 (about 9.49). Choice C halves 90 instead of taking a square root.
Question 3 of 3
Which expression is equal to √72?
Show answer and explanation
Answer: C.72 = 36 × 2, so √72 = √36 × √2, which is 6√2. Choice A forgets to take the square root of 36, B equals √128, and D equals √864.
Square Root: frequently asked questions
Can you take the square root of a negative number?
Not among the real numbers. Any real number times itself is zero or positive, so nothing squares to −25. In Algebra 2 you meet imaginary numbers, where √−1 is written i and √−25 = 5i. In middle school and on most placement tests, the answer is that the square root of a negative number is not a real number.
Why is the square root of 49 not -7?
Both 7 and −7 square to 49, so 49 has two square roots. The symbol √49, though, names only the principal square root, which is the non-negative one: 7. To mean the negative root, write −√49 = −7. When you solve x2 = 49, you need both: x = ± 7.
Is the square root of 2 a rational number?
No. √2 ≈ 1.41421356… is irrational: it cannot be written as a fraction of two integers, and its decimal never ends or repeats. The same is true for the square root of every whole number that is not a perfect square, such as √3, √5 and √50.
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