Radical Expressions • Topic 2 of 4

Operations with Radicals

You can add or subtract radicals only when they are "like" — the same root of the same number — by combining their coefficients: 3√2 + 5√2 = 8√2. Often you must simplify first so like radicals appear: √8 + √2 = 2√2 + √2 = 3√2. To multiply radicals, multiply the numbers under the roots (and any outside coefficients): √6·√3 = √18 = 3√2, and (2√3)(4√6) = 8√18 = 24√2. Squaring a square root removes it: (√5)² = 5. Treat radicals like variables when adding, but like numbers under multiplication.

✅ Solved examples

1. Simplify 3√5 + 2√5.
Like radicals: (3 + 2)√5 = 5√5.
2. Simplify √8 + √2.
√8 = 2√2, so 2√2 + √2 = 3√2.
3. Simplify √6 · √3.
√18 = √(9·2) = 3√2.
4. Simplify (√7)².
Squaring undoes the root: 7.

✏️ Practice — try these, take hints as needed

1. Simplify 4√3 + √3.
Like radicals.
(4 + 1)√3.
—
5√3.
2. Simplify √27 − √12.
√27 = 3√3, √12 = 2√3.
Subtract coefficients.
—
√3.
3. Simplify √2 · √8.
√(2·8) = √16.
Evaluate.
—
4.
4. Simplify (√10)².
Squaring removes the root.
—
—
10.
5. Simplify √50 + √18.
√50 = 5√2, √18 = 3√2.
Add coefficients.
—
8√2.

📝 Topic test — 8 questions

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