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Radical Form Calculator

Convert rational exponents to radical form and back, or simplify any radical to simplest radical form — with every step shown.

50√10

Simplest radical form
50√10
Perfect power extracted
2500
Decimal value
158.113883
Exponent form
25000^(1/2)

Factor tree

22255555

Highlighted primes form complete groups of 2 and come out of the root.

  1. Prime factorise: 25000 = 2³ × 5⁵.
  2. Group each prime into sets of 2; every complete set comes out of the root as one factor.
  3. Largest perfect square factor: 2500 = 50², leaving 10 inside.
  4. √25000 = √(2500 × 10) = 50√10.

Index, radicand and rational exponents

A radical ⁿ√a has an index n (the small number; a plain √ means n = 2) and a radicand a (everything under the root). A rational exponent packs both into one fraction: the denominator becomes the index and the numerator becomes the power.

xmn=n√xmdenominator → indexnumerator → powerradicand = xᵐ

x^(m/n) = ⁿ√(xᵐ) = (ⁿ√x)ᵐ

Worked examples

1. √25000 in simplest radical form

  1. Prime factorise: 25000 = 2³ × 5⁵.
  2. Group each prime into sets of 2; every complete set comes out of the root as one factor.
  3. Largest perfect square factor: 2500 = 50², leaving 10 inside.
  4. √25000 = √(2500 × 10) = 50√10.

Answer: 50√10 ≈ 158.1139.

2. 8^(2/3) as a radical, evaluated

  1. The denominator 3 becomes the index of the root; the numerator 2 becomes the power of the base.
  2. 8^(2/3) = ³√(8²).
  3. ³√64 simplifies to 4.
  4. Value: 4.

3. x^(3/4) in radical form

  1. The denominator 4 becomes the index of the root; the numerator 3 becomes the power of the base.
  2. x^(3/4) = ⁴√(x³).

Answer: ⁴√(x³), equivalently (⁴√x)³.

Common mistakes

  • Swapping index and power — the denominator is the index: 8^(2/3) is ³√(8²), not √(8³).
  • Not extracting the largest perfect square — √72 = 3√8 is not finished; 72 = 36 × 2 gives 6√2.
  • Adding unlike radicals — √2 + √3 is not √5. Only like radicals combine: 2√3 + 5√3 = 7√3.
  • Even roots of negative numbers — √(−9) has no real value; odd roots are fine: ³√(−8) = −2.

Frequently asked questions

What is radical form?

Radical form writes a power with a root symbol instead of a fractional exponent. x^(m/n) in radical form is ⁿ√(x^m): the root's index is n and the radicand is x raised to m.

How do you convert a fractional exponent to radical form?

Reduce the fraction, then make the denominator the index of the root and the numerator the power of the base. 8^(2/3) becomes ³√(8²), which equals ³√64 = 4.

What is simplest radical form?

A radical is in simplest form when the radicand has no perfect-power factor for that index left inside, there is no fraction under the root, and there is no root in a denominator. √50 = 5√2 is simplest form; √50 is not.

How do you write a square root in simplest radical form?

Prime factorise the number, pull out each pair of equal primes as one factor in front, and leave the unpaired primes inside. 25000 = 2³ × 5⁵, so √25000 = 2 × 5² × √(2 × 5) = 50√10.

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