Exponents & Roots • Topic 4 of 5

Simplifying Radicals

To simplify a square root, pull out the largest perfect-square factor: √72 = √(36·2) = 6√2. The reliable method is to factor the number, pair up matching primes, and move each pair outside the root — 72 = 2³·3² gives one pair of 2s and one pair of 3s, leaving √72 = 2·3·√2 = 6√2. Radicals multiply and divide freely (√a·√b = √(ab), √a/√b = √(a/b)) but do NOT add across a sum: √(9 + 16) = √25 = 5, which is not √9 + √16 = 7. You can combine only "like" radicals, the way you combine like terms: 3√2 + 5√2 = 8√2, but 3√2 + 5√3 stays as it is. The GRE also expects you to rationalise a denominator — clear a root from the bottom by multiplying top and bottom by that root: 1/√2 = √2/2. These moves keep answers in the exact form the GRE’s answer choices use, since the basic calculator can’t produce them.

✅ Solved examples

1. Simplify √50.
50 = 25·2, and 25 is a perfect square: √50 = √25·√2 = 5√2.
2. Simplify √12 + √27.
√12 = 2√3 and √27 = 3√3. Like radicals: 2√3 + 3√3 = 5√3.
3. Simplify √6 · √15.
√6·√15 = √90 = √(9·10) = 3√10.
4. Rationalise 6/√3.
Multiply top and bottom by √3: 6√3/3 = 2√3.

✏️ Practice — try these, take hints as needed

1. Simplify √98.
Find the largest square factor.
98 = 49·2.
√49 = 7.
7√2
2. Simplify √45 + √20.
√45 = 3√5, √20 = 2√5.
Like radicals add.
3√5 + 2√5.
5√5
3. Simplify √8 · √6.
Multiply under one root.
√48.
48 = 16·3.
4√3
4. Rationalise 10/√5.
Multiply by √5/√5.
10√5/5.
Simplify.
2√5
5. Simplify √(75/3).
Divide under the root first.
75/3 = 25.
√25.
5

📝 Topic test — 8 questions

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