The square root of a number is the value that, multiplied by itself, gives the original number back: √81 = 9 because 9 × 9 = 81. The cube root is the value that cubes to the original: ∛27 = 3 because 3 × 3 × 3 = 27. Roots are the inverse of powers, and thinking of them that way makes them easy to compute and easy to check.
Key Takeaways
√xanswers “what squares to x?” and∛xanswers “what cubes to x?”.√81 = 9,∛64 = 4, and∛(-8) = -2, so cube roots of negatives exist, square roots of negatives do not (in real arithmetic).- Roots are fractional powers:
√x = x^(1/2),∛x = x^(1/3),nthRoot(x, n) = x^(1/n).- Whole-number results come from perfect squares and cubes; everything else is an irrational decimal.
What is a square root?
The square root of a number x, written √x, is the nonnegative number whose square equals x. Asking for √81 is the same as asking “what number, squared, gives 81?”, and the answer is 9.
Worked example 1: perfect squares. √81 = 9 because 9² = 81. √144 = 12 because 12² = 144. √0.25 = 0.5 because 0.5² = 0.25. Notice the pattern in the reverse direction: 9² = 81, then √81 = 9, so squaring and square-rooting cancel each other out.
The perfect squares worth memorizing
The numbers that come from squaring an integer are called perfect squares:
| Square | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Root | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
Any value inside this table collapses to a clean integer, which is a handy self-check: if you computed √64 and got anything other than 8, retrace your entry.
What is a cube root?
The cube root of a number x, written ∛x, is the number whose cube equals x. Because a negative times a negative times a negative is still negative, cube roots of negative numbers are perfectly valid and negative: ∛(-8) = -2.
Worked example 2: perfect cubes. ∛27 = 3 because 3³ = 27. ∛64 = 4 because 4³ = 64. ∛512 = 8 because 8³ = 512, and 8 × 8 × 8 = 512 checks out on any calculator.
The perfect cubes worth knowing
| Cube | 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729 | 1000 |
|---|---|---|---|---|---|---|---|---|---|---|
| Root | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
Why most square roots are irrational
Unless the number under the root is a perfect square or cube, the result never terminates. √2 cannot be written as a fraction of whole numbers, so it is irrational: the decimal goes on forever without repeating. A calculator cannot print an infinite decimal, so it shows a rounded approximation instead.
Worked example 3: irrational roots. √2 displays as 1.41421356237 on the calculator, √3 as 1.73205080757, and √5 as 2.2360679775. Guessing at these by hand is hopeless, which is why the root keys exist. The values are also a good reminder that the displayed decimal is rounded, and the floating-point arithmetic guide explains how small rounding errors can sneak into results.
Roots as fractional powers
Every root is also a power: the square root is the one-half power, the cube root is the one-third power, and so on. √x = x^(1/2) and ∛x = x^(1/3).
Worked example 4: roots as powers. 8^(1/3) = ∛8 = 2. 9^(1/2) = √9 = 3. 16^(1/4) = 2, since 2⁴ = 16. This “root = reciprocal exponent” link is the same idea the exponents explained post uses for negative and fractional powers, and it is the fastest way to type an unusual root on the calculator: instead of a dedicated key, just raise the number to a fraction.
Finding the nth root with the calculator
The calculator provides a general nthRoot(x, n) function that finds any root at all: nthRoot(x, n) is the number that raised to the nth power gives x. It takes two arguments separated by a comma, and it is documented alongside every other function in the calculator function reference.
Worked example 5: nth roots. nthRoot(81, 4) = 3, since 3⁴ = 81. nthRoot(32, 5) = 2, since 2⁵ = 32. nthRoot(0.001, 3) = 0.1, since 0.1³ = 0.001. When the index and the number are both chosen from perfect powers, the whole expression collapses to a clean integer.
Square and cube roots in the real world
Roots show up wherever a quantity grows by squares or cubes. The screen diagonal of a television is √(width² + height²); the side length of a square of area 144 cm² is √144 = 12 cm; a cube with volume 1000 cm³ has an edge of ∛1000 = 10 cm; and physics formulas for distance and energy are full of squares that get undone by roots. The exponent calculator is the right tool when your numbers are expressible as powers, because it keeps the working visible, while the scientific notation calculator formats the results that come back huge or tiny.
Frequently asked questions
What is the square root of a negative number?
There is no real-number answer, because a real number squared is never negative. The calculator flags √(-1) as an error, which is correct: the answer would need imaginary numbers (√(-1) = i). Cube roots of negative numbers are fine, though: ∛(-8) = -2.
What is the difference between √ and ∛?
√ undoes squaring and ∛ undoes cubing. √81 = 9 because 9² = 81, while ∛27 = 3 because 3³ = 27. For any other index, use nthRoot(x, n).
How do I calculate a root on the calculator?
Press √ for the square root, ∛ for the cube root, or type nthRoot(x, n) for any other index. You can also use fractional powers: 27^(1/3) gives the same result as ∛27.
Why does √2 give a long decimal instead of an exact answer?
Because √2 is irrational: it cannot be written as a fraction and its decimal never ends. The calculator shows 1.41421356237, a 12-significant-digit approximation. For exact engineering work, keep the root symbolic and only evaluate at the very end.