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Square Root Calculator
Calculate a square root, cube root, nth root, or integer power, with a simplified radical and a rounded decimal.
The number under the radical, such as 72 or 144
Enter a radicand to see the square root.
Enter a radicand to see the square root.
This square root calculator is for algebra class and homework that asks for a root or an integer power. Enter a radicand, or a base and an exponent, and the page shows the expression with those numbers, then the value. Whole-number radicands are reduced to simplified radical form, and a long decimal is rounded to eight significant figures.
Simplifying a square root
A square root is simplified when the number under the radical has no perfect-square factor other than 1. Split the radicand into its largest perfect square times whatever is left, then take the square root of that perfect square. 72 = 36 × 2, and the square root of 36 is 6, so √72 = 6√2. 144 is a perfect square, so √144 = 12. The square root of a negative number is not a real number.
Cube roots work the same way with groups of three equal factors, and a negative radicand is allowed. The cube root calculator covers perfect cubes, simplified cube roots, and negative cube roots in detail.
Higher roots and integer powers
The nth root mode takes a whole-number index of 2 or more. An even root of a negative number is not a real number. An odd root keeps the sign, the same way a cube root does. The exponent mode evaluates a whole-number power, so 2¹⁰ = 1024. A negative exponent is a reciprocal: 5⁻² = 1/25, and 2⁻³ = 1/8. Zero to the power of zero is undefined, and zero to a negative power is undefined because that would divide by zero. When a decimal runs on, the display keeps 8 significant figures and says that it is rounded.
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Frequently Asked Questions
How do you simplify the square root of 72?
Write 72 as 36 × 2. The square root of 36 is 6, and 2 has no perfect-square factor, so √72 = 6√2. The decimal is 8.4852814, rounded to 8 significant figures.
What is the square root of a negative number?
The square root of a negative number is not a real number. No real number multiplied by itself is negative, so the calculator explains that in a sentence.
Can this calculator find a cube root?
Yes. The nth root mode with index 3 gives a cube root, so ∛54 = ∛(27 × 2) = 3∛2. For perfect cubes, negative cube roots, and step-by-step simplifying, use the cube root calculator.
What does a negative exponent mean?
A negative exponent means a reciprocal. 2⁻³ = 1/2³ = 1/8, and 5⁻² = 1/25, which is 0.04 as a decimal. The exponent in this tool is a whole number.
Why is zero to the power of zero undefined?
Zero to the power of zero has no agreed value, so the calculator does not invent one. Zero raised to a negative power is also undefined, because it would divide by zero. Any non-zero number to the power of 0 is 1.