Guides / Radicals

Simplifying square roots and rationalizing the denominator

A square root splits over multiplication and never over addition. Almost every rule for simplifying radicals, and almost every mistake, comes down to which of those two you're looking at.

Why it works

For a, b ≥ 0, √ab = √a · √b. The reason: (√a · √b)² = ab, and √a · √b isn't negative, so it is the square root of ab. That's the only splitting rule. There is none for sums: (√a + √b)² = a + 2√ab + b, which has an extra middle term, so √a + √b is not the square root of a + b.

Rationalizing never changes a value. Multiplying by √3√3 is multiplying by 1, and the bottom becomes √3 · √3 = 3. For a binomial bottom, the conjugate works because of the difference of squares: (a − √b)(a + √b) = a² − b. The two middle terms cancel, and the root gets squared away.

Finally, √ always gives the non-negative root. If x = −3, then √x² = √9 = 3, which is |x|, not x.

Example 1: the largest perfect-square factor

Simplify √72.

1. √36 · 2The perfect squares are 4, 9, 16, 25, 36, 49, … The largest one that divides 72 is 36.
2. √36 · √2A root splits over multiplication.
3. 6√2√36 = 6. Nothing under the root has a square factor left, so it's fully simplified.

Starting with a smaller square still works, it just takes more lines: √72 = √9 · 8 = 3√8, and 8 = 4 · 2 still has a square in it, so 3√8 = 3 · 2√2 = 6√2. Stopping at 3√8 is correct in value but not simplified. Variables work the same way: for x ≥ 0, √50x³ = √25x² · 2x = 5x√2x.

Example 2: combining like radicals

Simplify √50 + √18 − √8.

1. √25 · 2 + √9 · 2 − √4 · 2As written, these are unlike radicals and can't be combined. Simplify each one first.
2. 5√2 + 3√2 − 2√2Take the root of each perfect square.
3. 6√2Now all three are multiples of √2, so add the coefficients like 5x + 3x − 2x: 5 + 3 − 2 = 6.

Example 3: rationalizing a single root

Rationalize 6√3.

1. 6√3 · √3√3Multiply by √3/√3, which equals 1. Both top and bottom, or the value changes.
2. 6√33√3 · √3 = 3, so the bottom is now a whole number.
3. 2√36 ÷ 3 = 2. Divide the 6 by 3, not the 3 under the root.

Example 4: rationalizing with the conjugate

Rationalize 43 − √5.

1. 43 − √5 · 3 + √53 + √5The conjugate of 3 − √5 is 3 + √5: same terms, opposite sign in the middle.
2. 4(3 + √5)9 − 5Bottom: (3 − √5)(3 + √5) = 3² − (√5)² = 9 − 5. The ±3√5 cross terms cancel.
3. 4(3 + √5)49 − 5 = 4.
4. 3 + √5The 4s cancel because 4 is a factor of the whole numerator.

Check with a calculator: 4 ÷ (3 − 2.236) ≈ 5.236 and 3 + 2.236 = 5.236.

Common mistakes, and the exact line they happen on

1. Splitting a root over a sum

1. √x² + 9 = x + 3This is the mistake. A root splits over multiplication, not addition. At x = 4: √(16 + 9) = √25 = 5, but 4 + 3 = 7.

√x² + 9 doesn't simplify at all. It's the square-root version of writing (a + b)² = a² + b².

2. Adding unlike radicals

1. √2 + √3 = √5This is the mistake. √2 + √3 ≈ 1.414 + 1.732 = 3.146, but √5 ≈ 2.236. Different numbers under the root means unlike radicals, and the sum stays as √2 + √3.

Like radicals behave like like terms: 3√2 + 2√2 = 5√2, the same way 3x + 2x = 5x. The number under the root stays put. It does not become √4.

3. Pulling out the square instead of its root

1. √72 = √36 · 2Fine so far.
2. 36√2This is the mistake. The 36 was under the root, so what comes out is √36 = 6. 36√2 ≈ 50.9, but √72 ≈ 8.49.

4. Squaring the binomial denominator as if it had no middle term

Rationalize 43 − √5 (Example 4).

1. 43 − √5 · 3 − √53 − √5Allowed, since this is multiplying by 1, but it's the wrong factor.
2. 4(3 − √5)9 − 5This is the mistake. (3 − √5)² = 9 − 6√5 + 5 = 14 − 6√5, still irrational. Writing 9 − 5 leads to 3 − √5 ≈ 0.764, but the original is about 5.236.

Only the conjugate, with the opposite middle sign, makes the cross terms cancel.

5. Writing √x² = x when x could be negative

1. √x² − 10x + 25 = √(x − 5)²Factor the perfect square trinomial. Fine so far.
2. x − 5This is the mistake. At x = 1: √16 = 4, but 1 − 5 = −4. The root is never negative. The correct answer is |x − 5|.

The bars are only safe to drop when the problem tells you the expression inside isn't negative, as in √50x³ in Example 1, where x ≥ 0.

Practice

1. Simplify √180.

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   The largest perfect square dividing 180 is 36: √36 · 5 = 6√5.

2. Simplify 2√12 + √27 − √75.

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   2 · 2√3 + 3√3 − 5√3 = 4√3 + 3√3 − 5√3 = 2√3.

3. Rationalize 10√5.

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   Multiply by √5/√5: 10√5/5 = 2√5.

4. Rationalize 6√7 + 2.

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   Multiply by the conjugate (√7 − 2)/(√7 − 2). The bottom is 7 − 4 = 3, so you get 6(√7 − 2)/3 = 2(√7 − 2) = 2√7 − 4.

5. Simplify √18x², where x can be any real number.

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   √9x² · 2 = 3|x|√2. The absolute value is needed because x might be negative.