Exponents & Roots Solver
Radical Calculator & Exponent Solver
Exponents tell you how many times a number is multiplied by itself. Roots ask the opposite question — what number, when raised to a power, gives this result. This solver doubles as a radical calculator: enter any expression with √, ³√ or ⁿ√ and you get the simplified form with full working shown.
How to use the radical calculator
- Type
\sqrt{72}to simplify √72 → 6√2 - Type
\sqrt[3]{27}for the cube root → 3 - Type
2^{10}for 2 to the power of 10 → 1024 - Type
\sqrt{x^2 + 4x + 4}for symbolic simplification
The seven laws of exponents
$a^m \cdot a^n = a^{m+n}$
$a^m / a^n = a^{m-n}$
$(a^m)^n = a^{mn}$
$a^0 = 1$ (for a ≠ 0)
$a^{-n} = 1/a^n$
$a^{m/n} = \sqrt[n]{a^m}$
Simplifying radicals — how it works
Find the largest perfect-square (or perfect-cube, etc.) factor of the radicand, then split the radical. Example: √72 = √(36 × 2) = √36 × √2 = 6√2. Common simplifications:
- √48 = √(16 × 3) = 4√3
- √200 = √(100 × 2) = 10√2
- √300 = √(100 × 3) = 10√3
- √500 = √(100 × 5) = 10√5
- ³√54 = ³√(27 × 2) = 3 ³√2
Rationalising denominators
A fraction with a radical in the denominator (like 1/√2) is conventionally rewritten with the radical on top. Multiply numerator and denominator by the radical:
For a binomial denominator like (1 + √3), multiply by the conjugate (1 − √3) to remove the radical entirely.
Common square roots and cube roots
- √2 ≈ 1.41421, √3 ≈ 1.73205, √5 ≈ 2.23607, √7 ≈ 2.64575, √11 ≈ 3.31662
- ³√2 ≈ 1.25992, ³√3 ≈ 1.44225, ³√7 ≈ 1.91293, ³√10 ≈ 2.15443