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Square Root Calculator

Find the square root, cube root or any nth root of a number, with the simplified radical form, imaginary square roots of negatives and full working.

Level: Grades 8–9

  • Common Core standard 8.EE.A.2Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
  • Common Core standard HSN.RN.A.2Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Printable worksheet

Square Root Calculator

Know the answer you need? Pick an input to work backward.

Inputs
Root

Negative numbers work for odd roots only.

Result

Square root

8.485281

The square root of 72 is 8.4852813742 (6√2 in simplified radical form).

Simplified radical form
6√2
Both square roots (solutions of r² = 72)
±8.485281
Perfect square?
No
Nearest perfect squares
64 and 81 (√ between 8 and 9)
Root degree
2

See the step-by-step solution ↓

Step-by-step solution

  1. We're looking for the square root of 72: a number, call it r, where multiplying 2 copies of r together gives 72

    r2=72

  2. Estimate first: 72 lies between the perfect squares 64 = 8² and 81 = 9², so its square root is between 8 and 9

    64<72<81

  3. To get the exact form, find the largest perfect square that divides 72. It's 36 = 6², so 6 comes out in front of the root sign

    72=36×2=6√2

  4. Work out the square root as a decimal

    72=8.485281

  5. Check the answer: multiplying 2 copies of it together should give back the original number (give or take rounding)

    8.4852812≈72

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

How to solve this on a calculator

The keys to press to get this page's worked-example answer on an ordinary calculator, with what the display should show after each step. A named-model tab follows that calculator's official guidebook.

√72 on a scientific calculator

Problem
Find the square root of 72 as a decimal.
Formula
r = √x, the positive number with r × r = x
Example values
x = 72 (the number). These are the fixed values from the worked example on this page, so the keys stay the same whatever you type into the calculator above.
Why a scientific calculator
Square roots need a √ key, which a scientific calculator has and a basic one here doesn't. It also works on a statistics calculator (not on this site yet) or graphing calculator.
The whole sequence
√72)=x²=

Step by step

  1. √

    Press √. The calculator opens a bracket for the number.

    Display: √(

  2. 72

    Type the number, 72.

    Display: √(72

  3. )

    Close the bracket.

    Display: √(72)

  4. =

    Press =. √72 is about 8.485, between 8 (√64) and 9 (√81).

    Display: 8.4852813742

  5. x²=

    Check it: press x² and = to square the answer. You get back 72.

    Display: 72

If your answer looks wrong

  • If you get 36, you divided by 2. A square root is the number that multiplies by itself to give 72.
  • The decimal never ends, so the display is rounded. The exact form is 6√2, shown by the calculator above.
  • A negative number under √ gives an error: real square roots of negative numbers don't exist.

How this is checked: an automated test presses these keys on this site's scientific calculator, checks every display along the way, and confirms the answer matches the calculator on this page (last checked 2026-10-19). It has not been checked on any particular brand or model, and real calculators label some keys differently.

What this calculator does

Find the square root, cube root or any other root of a number, with every step shown. A root undoes a power. The square root of 72 is the number that, multiplied by itself, gives 72. The cube root of 72 is the number that gives 72 when you multiply three copies of it together. This calculator handles roots of any whole-number degree from 2 to 1,000.

Along with the decimal answer, it gives the exact answer for whole numbers in simplified radical form. That's the tidy form algebra and geometry classes ask for: √72 becomes 6√2, and ∛72 becomes 2∛9. (A radical is the root sign, √.)

For square roots, it also tells you whether the number is a perfect square, like 36 or 49. It shows which two perfect squares your number sits between, a handy way to estimate. And it lists both the positive and negative roots. The square root of a negative number is given as an imaginary number, such as √−72 = 6i√2.

Where it's used

Why this math matters outside the classroom:

  • Square rooms. A square room with 196 square feet of floor is √196 = 14 ft on each side.
  • Cube boxes. A cube-shaped box holding 27,000 cm³ has edges of ∛27,000 = 30 cm.
  • Exact lengths. Geometry students simplify √50 to 5√2 when giving an exact diagonal length.
  • Falling objects. An object dropped from 20 m hits the ground after √(2 × 20 ÷ 9.81), about 2.02 seconds.
  • Yearly growth. An investment that grew by a factor of 1.21 over 2 years grew √1.21 − 1 = 10% a year on average.

Grade level and standards: Grades 8–9; Common Core State Standards for Mathematics 8.EE.A.2, HSN.RN.A.2. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

  1. Choose square root, cube root or nth root. For an nth root, enter the degree n: 4 for a fourth root, 5 for a fifth root, and so on.
  2. Enter the number under the root sign. Decimals, exponent form such as 2e10, and negative numbers all work. A negative number gives an imaginary answer for a square root, and a negative answer for an odd root like a cube root.
  3. Read the decimal root and, for whole numbers, the simplified radical. The steps show how the number is broken down, plus a check that works back to the number you started with.

What the inputs mean

Number
The radicand: the number you're taking the root of. A negative number has an ordinary (real) root only when the degree is odd. Its square root is imaginary, and its 4th, 6th and other even roots aren't covered.
Root degree (n)
The degree, or index, of the root. The square root has degree 2 and the cube root degree 3. Any whole number from 2 to 1,000 works.

Formula and method

ⁿ√x = r  where  rⁿ = x
ⁿ√x = x^(1/n)
ⁿ√(aⁿ × b) = a × ⁿ√b
ⁿ√(−x) = −ⁿ√x  for odd n
√(−x) = √x × √(−1) = i√x  for x > 0

In the formulas, x is your number, n is the degree of the root and r is the answer. The small raised n in rⁿ means "n copies of r multiplied together". So the nth root of x is the number r where rⁿ equals x. That's the same as raising x to the power 1/n. In the third line, a and b are whole numbers: a is what comes out in front, and b is what stays under the root sign.

To simplify, the calculator breaks x into prime numbers (numbers like 2, 3 and 5 that can't be split further). For example, 72 = 2 × 2 × 2 × 3 × 3. It groups matching primes into sets of n. One copy of each full set moves out in front of the root sign. Whatever is left stays inside. Nothing left inside can be pulled out, so the result is in simplest form.

An odd power keeps the sign of the number: (−3)³ = −27. So every negative number has exactly one real odd root. An even power is never negative, which is why √(−4) has no ordinary answer. For square roots, the calculator uses the imaginary unit i, which stands for √−1. So √(−4) = √4 × i = 2i. Higher even roots of negatives have complex answers that this tool doesn't give.

Worked example

Simplify √72 and find its decimal value.

  1. Estimate: 72 lies between 64 = 8² and 81 = 9², so √72 is between 8 and 9.
  2. Break 72 into primes: 72 = 2³ × 3². The largest perfect square inside it is 2² × 3² = 36.
  3. Split the root: √72 = √36 × √2 = 6√2.
  4. As a decimal, √72 = 8.485281 (to six places).
  5. Check: 8.485281² ≈ 72.

Result: √72 = 6√2 ≈ 8.485281, and both ±8.485281 square to 72. The cube root of the same number is 2∛9 ≈ 4.160168, because 72 = 8 × 9 and 8 is the perfect cube 2³.

Understanding the result

The decimal root is exact for perfect powers, like √144 = 12. Otherwise it's irrational, meaning the digits never end or repeat. The simplified radical is the exact answer. That's why textbooks and exams prefer 6√2 to 8.485.

If the simplified form is still a plain root, such as √7, that's fine. It means no perfect square bigger than 1 divides the number, so there's nothing to pull out.

Assumptions and limitations

  • The main answer is the principal square root, the one that isn't negative. The equation r² = 72 also has the answer −6√2, so both are listed as ±8.485281.
  • For a negative number, the main square root is i times the square root of its size. That's the usual convention.
  • Simplified radical form is given for whole numbers up to 10¹² (one trillion). Decimals and larger numbers show the decimal root only.
  • Decimal results carry about 15–16 significant digits, the usual limit for computer math. So a check like 8.485281² may land a tiny bit away from 72 in the last digit.

Frequently asked questions

How do I simplify a square root by hand?

Look for the largest perfect square that divides the number: 4, 9, 16, 25, 36 and so on. Write the number as that square times what's left. Then take the square root of the perfect square and put it in front. For 50: 50 = 25 × 2, so √50 = 5√2.

Why can I take the cube root of a negative number but not the square root?

Multiply an odd number of negatives together and the answer stays negative. So (−3)³ = −27, and ∛(−27) = −3. Multiply an even number of them and the answer is always positive. So no ordinary number squares to −4.

What is the square root of a negative number?

It's an imaginary number. Mathematicians use i for the number whose square is −1. So √(−16) = √16 × √(−1) = 4i. Check: (4i)² = 16 × i² = 16 × (−1) = −16. Numbers that aren't perfect squares work the same way: √(−72) = 6i√2.

Does a square root have two answers?

Every positive number has two square roots, one positive and one negative. Both 6 and −6 square to 36. The √ symbol means the positive one, called the principal root. When you solve x² = 36, you write x = ±6 to include both.

Is a square root the same as raising to the power one half?

Yes. x^(1/2) means √x, and in general x^(1/n) is the nth root. That's why the steps for cube and higher roots are written as fraction powers, such as 72^(1/3).

How do I estimate a square root without a calculator?

Find the perfect squares on either side. 72 sits between 64 (8²) and 81 (9²). It's a little under halfway from 64 to 81, so √72 is a little under 8.5. The exact value is about 8.485.

How this calculator is checked

Method: Definition of the nth root (rⁿ = x); prime factorization for simplest radical form; imaginary unit i = √−1 for square roots of negatives. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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