How to Calculate an nth Root on a Simple Calculator

An nth root asks for the number that, multiplied by itself n times, gives the number under the root sign. For example, the cube root of 20 is the number whose cube is 20. The quickest method depends on whether your calculator has an exponent key or only basic arithmetic keys.

Use the reciprocal exponent

For a positive number A and a positive integer n, the nth root is written as $\sqrt[n]{A}$ and can be calculated as

$$\sqrt[n]{A}=A^{1/n}.$$

On a scientific calculator, use the power key—often labeled $x^y$, $y^x$, or ^—to enter the number raised to the reciprocal of the index. To find the cube root of 20, calculate $20^{1/3}$, which is approximately $2.7144$.

The order of key presses varies by calculator. A common sequence is to enter 20, press the power key, enter (1 ÷ 3), and press equals. Use parentheses around the fraction if the calculator supports them; otherwise, calculate $1/3$ first and then use that decimal as the exponent. Some calculators have a dedicated nth-root key, which may ask you to enter the index and then the number.

If your calculator has only basic keys

A calculator with only addition, subtraction, multiplication, and division can still approximate a root. Use Newton’s iteration. For a positive number A and positive integer n, choose a positive starting estimate $x_0$ and repeatedly calculate

$$x_{k+1}=\frac{(n-1)x_k+A/x_k^{n-1}}{n}.$$

Here, $x_k$ is your current estimate. The iteration converges to the positive root, although a poor starting estimate can make an intermediate value farther from the root. The power $x_k^{n-1}$ can be calculated by repeated multiplication. For instance, when finding a cube root, square the current estimate, divide A by that square, and then use the formula.

A sensible starting estimate helps. Look for nearby powers: if you are finding the cube root of 20, then $2^3=8$ and $3^3=27$, so start with $x_0=3$. Each iteration requires just multiplication, division, addition, and division by n.

Worked example: cube root of 20

For a cube root, $n=3$, so the iteration becomes

$$x_{k+1}=\frac{2x_k+20/x_k^2}{3}.$$

Start with $x_0=3$:

• First iteration: $x_1=(2\cdot3+20/3^2)/3\approx2.7407407$.
• Second iteration: $x_2=(2\cdot2.7407407+20/(2.7407407)^2)/3\approx2.7146696$.
• Third iteration: $x_3\approx2.7144177$.

Thus, the cube root of 20 is approximately $2.71442$. You can check it by cubing the estimate; the result is approximately 20. With a basic calculator, round each intermediate result to a few more decimal places than you need in the final answer.

For square roots, this method simplifies to the familiar averaging rule:

$$x_{k+1}=\frac{x_k+A/x_k}{2}.$$

For example, to find $\sqrt{10}$, start at 3: the next estimate is $(3+10/3)/2\approx3.1667$, and another iteration gives approximately $3.1623$.

Domain and sign

For a real-valued root, an even index requires a nonnegative number under the root sign. For example, $\sqrt[4]{-16}$ is not a real number. An odd index allows a negative radicand: $\sqrt[3]{-8}=-2$. On an exponent-key calculator, a negative base raised to a decimal approximation of a fraction may produce an error, even when the exact odd root is real. If that happens, calculate the root of the positive value and apply the negative sign afterward.

Common mistakes

• Using n instead of 1/n: To find a cube root, use the exponent $1/3$, not 3. Raising 20 to the power 3 gives 8000.
• Entering the expression in the wrong order: Power-key behavior differs across calculators. Check the manual or use parentheses so the entire fraction $1/n$ is the exponent.
• Rounding too early: Repeated rounding can reduce the accuracy of Newton’s iteration. Keep several decimal places during the steps, then round the final answer.
• Assuming the estimate is exact: An nth root may be irrational, so a decimal display is normally an approximation. Check by raising the result to the nth power and compare it with the original number.