Free Tool · Square & Nth Roots · Radical Simplification

Square Root Calculator

Compute square roots, cube roots, and any nth root — with the result shown in exact radical form when possible.

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Enter a whole number for the root index (2–10). Use 2 for square root, 3 for cube root, and so on. Radicand can be any real number.

What this square root calculator does

This tool computes the square root of any non-negative number and, more generally, the n-th root of any real number. For a radicand that is a perfect square (or perfect n-th power), it returns the exact integer or rational answer. For other radicands, it returns the decimal approximation and, where possible, the exact radical form in simplest terms. The step-by-step block shows the prime factorisation of the radicand and how the pairs of prime factors are pulled out of the radical.

Formulas

Square root definition
√x = r  if and only if  r2 = x  and  r ≥ 0
General nth root
√ x n = r  if and only if  rn = x
Product rule for radicals
√a × b = √a × √b
Quotient rule for radicals
√a / b = √a √b  (b ≠ 0)
Rationalising a denominator
k√a = k × √aa

Variable definitions

  • x — the radicand, the number under the radical sign.
  • n — the root index. n = 2 for square root, n = 3 for cube root, and so on.
  • r — the root, the value whose n-th power is x.

Worked examples

Example 1: √144 (perfect square)

  1. Prime factorisation: 144 = 24 × 32.
  2. Every prime appears an even number of times → 144 is a perfect square.
  3. Take one factor of each pair: 22 × 3 = 4 × 3 = 12.
  4. Result: √144 = 12

Example 2: √72 (simplify a radical)

  1. Prime factorisation: 72 = 23 × 32.
  2. Group pairs of primes: one pair of 2's and one pair of 3's. One 2 remains unpaired.
  3. Pull out the paired factors: 2 × 3 = 6. The unpaired 2 stays under the radical.
  4. Result: √72 = 6√2 ≈ 8.4852813742…

Example 3: ∛27 (cube root)

  1. Prime factorisation: 27 = 33.
  2. The exponent 3 matches the root index → perfect cube.
  3. Take one factor from the group: 3.
  4. Result: ∛27 = 3

Example 4: ∛(−8) (odd root of a negative number)

  1. The cube root of a negative number is defined because odd powers preserve sign.
  2. (−2)3 = −2 × −2 × −2 = −8.
  3. Result: ∛(−8) = −2

Example 5: √(−4) (even root of a negative — undefined in reals)

  1. Every real number squared is non-negative.
  2. There is no real r with r2 = −4.
  3. Result: no real square root exists. The complex roots would be ±2i.

Where square roots appear in Class 9 & 10

Square roots and radicals are central to several chapters of the BISE Punjab syllabus:

  • Quadratic formula — the discriminant sits under a square root.
  • Pythagoras theorem — finding the hypotenuse of a right triangle requires a square root.
  • Surds and indices — simplifying expressions like 6√2 + 3√2 and rationalising denominators.
  • Physics numericals — standard deviation, RMS values, and pendulum periods all use square roots.
  • Geometry — distance between two points is √((Δx)² + (Δy)²).

Common mistakes to avoid

  • Assuming √(a + b) = √a + √b. It is not. √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7. The product rule applies only to multiplication, not addition.
  • Forgetting the ± when solving x² = k. The equation x² = 9 has two solutions: x = +3 and x = −3. The symbol √9 by convention means only the positive root, but a full solution needs both.
  • Leaving a perfect square inside the radical. √72 should be simplified to 6√2. Leaving it as √72 loses marks in exams.
  • Not rationalising the denominator. An answer like 5/√3 is not in standard form. Multiply numerator and denominator by √3 to get 5√3/3.
  • Taking an even root of a negative number. √(−4) has no real solution. ∛(−8) does: it is −2. The rule is even index + negative radicand → no real root.
  • Confusing the root symbol with an exponent. √x is the same as x^(1/2), but writing it that way in an exam is not accepted if the question asks for a radical answer.

FAQ

What is a square root?

The square root of a non-negative number x is the number r such that r × r = x. Every positive number has two square roots, one positive and one negative, but the symbol √x by convention refers only to the non-negative root. For example, √9 = 3, and the two roots of 9 are 3 and −3.

Can the square root of a negative number be found?

No real number squared gives a negative. But a cube root of a negative number does exist, because a negative raised to an odd power is negative. For example, the cube root of −8 is −2 because (−2)³ = −8. This calculator rejects even roots of negative numbers and accepts odd roots.

How do I simplify a radical like √72?

Factor the number and pull out the largest perfect square. √72 = √(36 × 2) = √36 × √2 = 6√2. You can also factor into primes and group pairs: 72 = 2³ × 3², so √72 = 2 × 3 × √2 = 6√2.

What is the difference between a square root and an nth root?

A square root is a specific case of an nth root with n = 2. The nth root of x is the number that, when raised to the power n, gives x. For example, the cube root (n = 3) of 27 is 3 because 3³ = 27. This calculator supports n from 2 to 10.

How do I rationalise a denominator?

Multiply the numerator and denominator by the same radical that appears in the denominator. For example, 5/√3 = (5 × √3) / (√3 × √3) = 5√3 / 3. This removes the radical from the denominator, which is the standard form expected in exams.

Why do calculators give decimals when the answer is irrational?

Numbers like √2 are irrational — their decimal expansions never terminate and never repeat. Any calculator can only show an approximation. In exams, the exact form (like 6√2) is preferred over the decimal, because the decimal is always rounded.

Related tools

Related Hira Academy resources

Square roots and radicals appear throughout the BISE Punjab mathematics syllabus — from quadratic equations to surds. Revise the theory with our free Class 9 notes and Class 10 notes.

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