A radical is a root: √ for a square root, ∛ for a cube root. Simplifying one means pulling out every perfect square (or cube) so what is left under the sign is as small as possible. It is the same idea as reducing a fraction, and it shows up constantly once you start using the quadratic formula and the Pythagorean theorem.
Perfect squares to know
Simplifying is much faster when you recognize these on sight:
- Perfect squares (1² to 15²): 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.
- Perfect cubes (1³ to 10³): 1, 8, 27, 64, 125, 216, 343, 512, 729, 1,000.
Simplifying a square root
The key rule: √(ab) = √a × √b. Split the number into a perfect square times something else, then take the root of the square.
Example Simplify √72
- Find the largest perfect square that divides 72. Try 36: 72 = 36 × 2.
- √72 = √36 × √2 = 6√2.
If you don’t spot the largest square, you can still get there in stages: √72 = √(4 × 18) = 2√18 = 2√(9 × 2) = 2 × 3√2 = 6√2. Just keep going until no square factor is left.
Example Simplify 3√50
- 50 = 25 × 2, so √50 = 5√2.
- 3 × 5√2 = 15√2.
With variables
For square roots, each pair of a variable comes out as one. √(x⁶) = x³, because x³ × x³ = x⁶. An odd power leaves one behind.
Example Simplify √(48x⁵y²)
- Numbers: 48 = 16 × 3, so √48 = 4√3.
- x⁵ = x⁴ × x, so √(x⁵) = x²√x.
- √(y²) = y (assuming y is positive).
- Together: 4x²y√(3x).
Cube roots
Cube roots work the same way with perfect cubes, and groups of three.
Example Simplify ∛54 and ∛(16a⁷)
- 54 = 27 × 2, so ∛54 = 3∛2.
- 16 = 8 × 2 and a⁷ = a⁶ × a, so ∛(16a⁷) = 2a²∛(2a).
Unlike square roots, cube roots of negatives are fine: ∛(−27) = −3.
Adding and subtracting radicals
You can only combine like radicals, ones with the same number under the sign. Treat √2 like a variable: 3√2 + 5√2 = 8√2, just as 3x + 5x = 8x.
Simplify first. Radicals that look different often turn out to match.
Example Simplify √12 + √75 − √27
- √12 = 2√3, √75 = 5√3, √27 = 3√3.
- 2√3 + 5√3 − 3√3 = 4√3.
√a + √b is not √(a + b)
√9 + √16 = 3 + 4 = 7, but √25 = 5. Roots split over multiplication and division, never over addition or subtraction.
Multiplying radicals
Multiply the numbers outside together and the numbers inside together, then simplify.
Example Simplify 2√6 × 5√10
- Outside: 2 × 5 = 10. Inside: 6 × 10 = 60.
- 10√60 = 10 × 2√15 = 20√15.
Binomials with radicals expand like any other binomial. A useful fact: (√a)² = a.
Example Expand (3 + √5)(3 − √5)
3² − (√5)² = 9 − 5 = 4. The radicals cancel because this is a difference of squares, which is exactly the trick used below.
Rationalizing the denominator
Many teachers and tests want final answers without a radical in the denominator. To remove one, multiply the top and bottom by something that turns the denominator into a whole number.
Example Rationalize 6/√3
Multiply top and bottom by √3: 6√3/3 = 2√3.
When the denominator is a sum or difference, multiply by its conjugate, the same two terms with the opposite sign in the middle.
Example Rationalize 4/(1 + √2)
- Multiply top and bottom by 1 − √2.
- Bottom: (1 + √2)(1 − √2) = 1 − 2 = −1.
- Top: 4(1 − √2) = 4 − 4√2.
- Result: (4 − 4√2)/(−1) = 4√2 − 4.
Radicals as fractional exponents
A root is the same as a fractional exponent. The denominator is the root and the numerator is the power:
- √x = x1/2
- ∛x = x1/3
- ⁿ√(xm) = xm/n
So 82/3 = (∛8)² = 2² = 4. Writing roots as exponents lets you use every rule in the exponent rules guide, which is often the fastest way through a messy expression.
Solving radical equations
Isolate the radical, then raise both sides to the power that removes it. Always check your answers, because squaring both sides can create solutions that don’t work in the original equation. These are called extraneous solutions.
Example Solve √(2x + 3) = x
- Square both sides: 2x + 3 = x².
- Rearrange: x² − 2x − 3 = 0, which factors as (x − 3)(x + 1) = 0.
- Candidates: x = 3 or x = −1.
- Check x = 3: √9 = 3. ✓
- Check x = −1: √1 = 1, which is not −1. ✗
- The only solution is x = 3.
The factoring step is covered in factoring polynomials and solving quadratic equations.
Problems to try
Simplify √98.
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7√2. 98 = 49 × 2.
Simplify √(75a³b⁴).
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5ab²√(3a). 75 = 25 × 3, a³ = a² × a and √(b⁴) = b².
Simplify 2√20 − √45.
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√5. 2√20 = 4√5 and √45 = 3√5, so 4√5 − 3√5 = √5.
Simplify √8 × √18.
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12. √(8 × 18) = √144 = 12.
Rationalize 10/√5.
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2√5. Multiply by √5/√5 to get 10√5/5.
Evaluate 272/3.
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9. ∛27 = 3 and 3² = 9.
Solve √(x + 7) = x − 5.
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x = 9. Squaring gives x + 7 = x² − 10x + 25, so x² − 11x + 18 = 0 and x = 2 or x = 9. Checking x = 2 gives 3 = −3, which fails.