Root and Radical Calculator

What does the root calculator do?

It has three tabs. The root tab finds the square root, cube root or n-th root of a number, writes it in simplest radical form and gives the decimal value. The radical expression tab works out an expression with square roots exactly: it simplifies the roots, combines like radicals, multiplies and divides. The rationalise tab removes a root from the denominator by multiplying with the conjugate. Every tab shows its working step by step.

How is a root simplified?

The number under the root is factored into primes. For a square root, factors with exponent 2 (for a cube root, exponent 3) come out of the root. Example: 72 = 2³ · 3² = 36 · 2, so √72 = √36 · √2 = 6√2. For a cube root, 54 = 27 · 2 and ∛54 = 3∛2. In the root of a fraction the denominator is cleared: √(1/2) = √2/2.

Operations with radicals

• Adding and subtracting: only roots with the same radicand can be combined. In 3√8 + √50 − √2 we have √8 = 2√2 and √50 = 5√2, so the result is 6√2 + 5√2 − √2 = 10√2. On the other hand √2 + √3 cannot be simplified further.
• Multiplying: √a · √b = √(a·b). Example: √12 · √3 = √36 = 6.
• Squaring: (2 + √3)² = 4 + 4√3 + 3 = 7 + 4√3.

Type the root sign as √ or as sqrt(8). A coefficient goes in front: 3√8.

Rationalising the denominator

If the denominator is a single root, multiply the numerator and the denominator by that root: 1/√2 = √2/2. If it has two terms, multiply by its conjugate, because the roots disappear in (a + b)(a − b) = a² − b². Example: for 6/(√5 − 2) multiply top and bottom by (√5 + 2). The denominator becomes 5 − 4 = 1 and the numerator 6√5 + 12.

Limits

An even root of a negative number is not real and is not calculated. For simplification the number under the root (after clearing the denominator for fractions) can be at most 10¹⁵, and for larger numbers only the decimal value is given. The expression and rationalise tabs work with square roots only: cube roots, nested roots (such as √(2 + √3)), letters and π are not accepted. Decimal values are rounded to 12 significant digits, while the radical form is exact.