Radical Calculator

What is a radical?

A radical expression uses the root symbol √. In ⁿ√a, n is the index and a is the radicand. When no index is written, it is a square root (n = 2): √49 = 7 because 7² = 49. A cube root (n = 3) asks which number multiplied by itself three times gives the radicand: ∛125 = 5. For even indices, the radical sign means the principal (non-negative) root, so √49 is 7, not −7, even though (−7)² is also 49.

How to simplify a radical, step by step

1. Write the radicand as a product of prime factors: 72 = 2 × 2 × 2 × 3 × 3.
2. Group the factors into sets of n identical factors (pairs for square roots, triples for cube roots): (2 × 2) × (3 × 3) × 2.
3. Take one factor out of the root for each complete group: 2 × 3 = 6 outside.
4. Multiply what is left inside the root: 2.
5. Write the result: √72 = 6√2.

For ∛54: 54 = 2 × 3 × 3 × 3. One group of three 3s comes out as 3, leaving 2 inside, so ∛54 = 3∛2. A radical is in simplest form when the radicand has no perfect-power factors left, there are no fractions under the root and no roots in a denominator.

Shortcut: look for the largest perfect square

If you know your perfect squares, find the largest one that divides the radicand: 72 = 36 × 2, so √72 = √36 × √2 = 6√2. Using a smaller square works too but takes more steps – 72 = 4 × 18 gives 2√18, which still simplifies to 2 × 3√2 = 6√2.

Adding and subtracting radicals

Only like radicals – the same index and the same radicand – can be combined, just as you combine like terms in algebra. First simplify each radical, then add the coefficients. √50 + √18 = 5√2 + 3√2 = 8√2. But √2 + √3 cannot be combined; it stays as it is, about 3.146.

Multiplying and dividing radicals

Radicals with the same index can be multiplied or divided under one root: √6 × √15 = √90 = 3√10, and √72 ÷ √2 = √36 = 6. Multiply coefficients separately: 2√3 × 4√5 = 8√15. Radicals with different indices need to be rewritten with rational exponents and a common denominator first: √2 × ∛2 = 2^(1/2) × 2^(1/3) = 2^(5/6) = ⁶√32.

Rationalising the denominator

By convention, simplest form has no radical in the denominator. Multiply the top and bottom by whatever removes it:

• Single square root: 6 ÷ √3 = (6 × √3) ÷ (√3 × √3) = 6√3 ÷ 3 = 2√3.
• Cube root: 1 ÷ ∛2 = ∛4 ÷ ∛8 = ∛4 ÷ 2, because multiplying by ∛4 makes a perfect cube underneath.
• Binomial with roots: multiply by the conjugate. 1 ÷ (√5 − √2) = (√5 + √2) ÷ ((√5)² − (√2)²) = (√5 + √2) ÷ 3.

Radicals and rational exponents

Every radical can be written as a power: ⁿ√a = a^(1/n), and ⁿ√(aᵐ) = a^(m/n). This makes exponent rules available: 8^(2/3) = (∛8)² = 2² = 4, and 16^(3/4) = (⁴√16)³ = 2³ = 8. Negative exponents mean reciprocals, so 27^(−1/3) = 1 ÷ 3.

Roots of negative numbers

Odd roots of negative numbers are real: ∛(−27) = −3, because (−3)³ = −27. Even roots of negative numbers are not real numbers. In the complex numbers, √(−1) is defined as i, so √(−16) = √16 × √(−1) = 4i. The calculator shows this result and labels it as imaginary.

Simplifying radicals with variables

The same rule applies to variables: divide each exponent by the index. √(x⁶) = x³, √(x⁵) = x²√x and ∛(a⁷b³) = a²b∛a. For even roots, a strict answer uses absolute values where needed – √(x²) = |x| – unless the variable is known to be non-negative.

Common mistakes

• √(a + b) is not √a + √b: √(9 + 16) = √25 = 5, but √9 + √16 = 7.
• Combining unlike radicals: √2 + √3 ≠ √5.
• Forgetting to simplify fully: 2√18 should become 6√2.
• Dropping the index: the 3 in ∛ matters, and ∛8 = 2 while √8 = 2√2.
• Losing a negative sign with odd roots: ∛(−8) = −2, not 'undefined'.

Higher roots step by step

The same method works for any index. For the fourth root of 48, factor 48 = 2⁴ × 3; one group of four 2s comes out, so ⁴√48 = 2⁴√3. For the fifth root of 96, 96 = 2⁵ × 3, so ⁵√96 = 2⁵√3. For ∛250, 250 = 5³ × 2, so ∛250 = 5∛2. Whenever a prime appears fewer than n times, it stays under the root.

Fractions under a root

Split the root across the numerator and denominator, then simplify each part: √(8/9) = √8 ÷ √9 = 2√2 ÷ 3. If the denominator is not a perfect power, rationalise: √(5/2) = √5 ÷ √2 = √10 ÷ 2. Decimals work the same way once written as fractions – √0.5 = √(1/2) = √2 ÷ 2 ≈ 0.7071.

Multiplying binomials with radicals

Use the distributive property (FOIL) and simplify each term. (√2 + √3)² = 2 + 2√6 + 3 = 5 + 2√6. Conjugate pairs remove the radicals entirely: (2 + √3)(2 − √3) = 4 − 3 = 1. That is exactly why multiplying by a conjugate rationalises a denominator.

Estimating square roots without a calculator

Find the two perfect squares either side: √10 lies between √9 = 3 and √16 = 4, and much closer to 3. For more accuracy, use the Babylonian (Newton's) method – average your guess with the number divided by your guess, and repeat. Starting with 3: (3 + 10 ÷ 3) ÷ 2 ≈ 3.1667; then (3.1667 + 10 ÷ 3.1667) ÷ 2 ≈ 3.16228, already correct to five decimal places (√10 ≈ 3.162278). Each step roughly doubles the number of correct digits.

Solving radical equations

To solve an equation with a radical, isolate the radical, raise both sides to the power of the index, solve, and then check every answer in the original equation. Squaring can create 'extraneous' solutions that do not work. Example: √(x + 3) = x − 3. Squaring gives x + 3 = x² − 6x + 9, so x² − 7x + 6 = 0 and x = 6 or x = 1. Checking: x = 6 gives √9 = 3 ✓; x = 1 gives √4 = 2, but 1 − 3 = −2 ✗. The only solution is x = 6.

Radicals in geometry

• Diagonal of a square with side s: s√2. A 5 cm square has a diagonal of 5√2 ≈ 7.07 cm.
• Height of an equilateral triangle with side s: (s√3) ÷ 2. For s = 6, the height is 3√3 ≈ 5.20.
• Distance between two points: √((x₂ − x₁)² + (y₂ − y₁)²). From (1, 2) to (4, 6) the distance is √(9 + 16) = 5.
• Space diagonal of a cube with side s: s√3.

Exact radical answers are preferred in maths classes because they keep full precision; decimal approximations are fine for measurements in the real world.

A note on the radical symbol

The word radical comes from the Latin radix, meaning root. The √ sign appeared in print in 16th-century German algebra books, and the bar over the radicand (the vinculum) was added later so that it was clear how much of an expression is under the root – which is why √(a + b) and √a + b mean different things.

Practice problems with answers

Work each one by hand using prime factors, then check it in the calculator. If your answer differs, compare the factorisation steps – most errors come from missing a perfect-square factor or making an arithmetic slip while multiplying the factors that remain under the root.

Cube roots of common numbers

Recognising perfect cubes speeds up simplification: 8, 27, 64, 125, 216, 343, 512, 729 and 1,000 are the cubes of 2 to 10. So ∛16 = ∛(8 × 2) = 2∛2, ∛81 = ∛(27 × 3) = 3∛3 and ∛192 = ∛(64 × 3) = 4∛3. Negative cubes work too: ∛(−125) = −5.

Where radicals show up outside the classroom

Square roots appear whenever lengths, areas and squared quantities meet. The standard deviation in statistics is the square root of the variance; the distance a falling object covers relates to the square root of time; the side of a square room with an area of 20 m² is √20 = 2√5 ≈ 4.47 m; and the 'rule of 72'-style shortcuts in finance give way to roots when you work out a constant growth rate – the average annual growth that doubles money in 8 years is ⁸√2 − 1 ≈ 9.05%. In each case the calculator gives the exact form and the decimal you would use in practice.