Find the square root, cube root or any other root of a number, with the exact simplified form when there is one.
Find √x for a number x.
Find ∛x for a number x. Negative numbers have a real cube root.
Find the nth root of a number, ⁿ√x.
The nth root of a number a is the number b that gives a when it is multiplied by itself n times, so bⁿ = a. The square root, where n is 2, is the number that gives the original when squared, so the square root of 16 is 4. The cube root, where n is 3, is the number that gives it when cubed, so the cube root of 27 is 3. Roots are the opposite of powers, and a root can also be written as a fractional exponent: the nth root of a is a to the power 1 ÷ n.
A positive number has a positive and a negative square root, for example both 4 and −4 square to 16, and the calculator shows the positive one, called the principal root. The square root of a negative number is not a real number. It is written with the imaginary unit i, so the square root of −4 is 2i. Odd roots, such as the cube root, do exist for negative numbers: the cube root of −8 is −2. Even roots of negative numbers have no real answer.
Many roots are not whole numbers, and their decimals go on forever. When the number under the root has a factor that is a perfect power, it can be taken out, which gives a neat exact answer. The square root of 50 equals 5 times the square root of 2, because 50 is 25 × 2 and 25 is a perfect square. The calculator shows this exact form where it applies, along with a decimal value rounded to about 13 decimal places.
Find the number that gives the original when multiplied by itself. For perfect squares, such as 16 or 81, this is a whole number. Others, such as 50, give a long decimal.
Not in real numbers. The result is an imaginary number written with i. The square root of −9 is 3i.
It is the number that gives the original when multiplied by itself three times. The cube root of 64 is 4 because 4 × 4 × 4 is 64.
Raise the number to the power 1 ÷ n. The 4th root of 16 is 16 to the power 0.25, which is 2.
Split the number into a perfect square times another number, then take the root of the perfect square out. For 72, which is 36 × 2, the square root is 6 times the square root of 2.
Pick another tool to jump straight to it.