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Roots Calculator.

Calculate square roots, cube roots, and nth roots with step-by-step simplification.

x\sqrt{x}

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Enter a number to calculate its root with step-by-step simplification

Understanding Roots and Radicals

A root (or radical) is the inverse operation of exponentiation. If bn=ab^n = a, then b is the nth root of a, written as an\sqrt[n]{a}. The most common roots are square roots (n=2) and cube roots (n=3).

\sqrt{} Square Root

a=b\sqrt{a} = b means b2=ab^2 = a

16=4\sqrt{16} = 4
because 42=164^2 = 16

3\sqrt[3]{} Cube Root

a3=b\sqrt[3]{a} = b means b3=ab^3 = a

273=3\sqrt[3]{27} = 3
because 33=273^3 = 27

n\sqrt[n]{} Nth Root

an=b\sqrt[n]{a} = b means bn=ab^n = a

814=3\sqrt[4]{81} = 3
because 34=813^4 = 81

How to Simplify Square Roots

Simplifying radicals involves finding the largest perfect square factor and extracting it from under the radical sign.

Example: Simplify 72\sqrt{72}

Step 1: Find prime factorization

72=23×32=8×972 = 2^3 \times 3^2 = 8 \times 9

Step 2: Identify perfect square factors

72=36×272 = 36 \times 2 (36 is a perfect square)

Step 3: Apply the product rule ab=a×b\sqrt{ab} = \sqrt{a} \times \sqrt{b}

72=36×2=62\sqrt{72} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}

Result: 72=62≈8.485\sqrt{72} = 6\sqrt{2} \approx 8.485

Perfect Squares and Perfect Cubes

Perfect Squares (n2n^2)

1
121^2
4
222^2
9
323^2
16
424^2
25
525^2
36
626^2
49
727^2
64
828^2
81
929^2
100
10210^2

Perfect Cubes (n3n^3)

1
131^3
8
232^3
27
333^3
64
434^3
125
535^3
216
636^3
343
737^3
512
838^3
729
939^3
1000
10310^3

Properties of Roots

PropertyFormulaExample
Product Rulea×b=a×b\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}4×9=4×9=2×3=6\sqrt{4 \times 9} = \sqrt{4} \times \sqrt{9} = 2 \times 3 = 6
Quotient Ruleab=ab\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}164=164=42=2\sqrt{\frac{16}{4}} = \frac{\sqrt{16}}{\sqrt{4}} = \frac{4}{2} = 2
Power Ruleamn=am/n\sqrt[n]{a^m} = a^{m/n}82=82/2=8\sqrt{8^2} = 8^{2/2} = 8
Nested Rootsamn=an×m\sqrt[n]{\sqrt[m]{a}} = \sqrt[n \times m]{a}16=164=2\sqrt{\sqrt{16}} = \sqrt[4]{16} = 2
Rationalization1a=aa\frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a}12=22\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}

Real-World Applications of Roots

Distance Formula

Finding distance between two points uses the square root: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

Pythagorean Theorem

Finding the hypotenuse of a right triangle: c=a2+b2c = \sqrt{a^2 + b^2}

Standard Deviation

Statistics uses square root to calculate spread: σ=Σ(x−μ)2n\sigma = \sqrt{\frac{\Sigma(x-\mu)^2}{n}}

Volume to Side Length

Finding cube edge from volume: side=Volume3\text{side} = \sqrt[3]{\text{Volume}}

Compound Interest

Finding annual rate from total growth uses nth roots

Physics & Engineering

Wave calculations, electrical circuits, and structural analysis

Negative Numbers and Complex Roots

Even roots (square root, 4th root, etc.) of negative numbers are not real numbers. They produce complex numbers involving i, where i=−1i = \sqrt{-1}.

−16=4i\sqrt{-16} = 4i (not a real number)
−814=3i\sqrt[4]{-81} = 3i (not a real number)

Odd roots (cube root, 5th root, etc.) of negative numbers are real negative numbers.

−273=−3\sqrt[3]{-27} = -3 (because (−3)3=−27(-3)^3 = -27)
−325=−2\sqrt[5]{-32} = -2 (because (−2)5=−32(-2)^5 = -32)

Pro Tip for Students

When simplifying radicals, always look for the largest perfect square (or cube) factor first. This gives you the most simplified form in fewer steps. For example, 200=100×2=102\sqrt{200} = \sqrt{100 \times 2} = 10\sqrt{2} is faster than finding 4×50=250\sqrt{4 \times 50} = 2\sqrt{50} and then simplifying again.

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