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By Macrobert Thomas M

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Simplify. Example 2 Using the Product Rule to Simplify Square Roots Simplify the radical expression. — — a.  ​ √ 162a5b4 ​ Solution — a. √ ​ 100 ∙ 3 ​ — — ​ 100 ​∙ √ ​ 3 ​ √ — Factor perfect square from radicand. Write radical expression as product of radical expressions. 10​√ 3 ​ Simplify. 3 Radicals and Rational Expressions — b. ​√81a4b4 ∙ 2a ​ — — ​√81a b ​ ∙√ ​ 2a ​ 4 4 Factor perfect square from radicand. Write radical expression as product of radical expressions. — 9a b ​√2a ​ Simplify. ​ How To… Given the product of multiple radical expressions, use the product rule to combine them into one radical expression.

3 Radicals and Rational Expressions Solution Rewrite each term so they have equal radicands. — — — — — — — 2 — = 20​√9 ​​√ 4 ​​√ 2 ​​√ a √ ​​ a2 ​ √ ​ (​ b2)​​ ​​√ c ​ 20​ √ 72a3b4c ​ — ​ = 20(3)(2)​∣ a ∣​b2​√2ac ​ = 120​∣ a ∣​b2​√2ac — — — — — — — 2 — ​ = 14​√2 ​​ √4 ​​ √a ​​ √a2 ​ √ ​ (​ b2)​​ ​​ 14​√8a3b4c √ c ​ — ​ = 14(2)​∣ a ∣​b2​√2ac ​ = 28​∣ a ∣​b2​√2ac — Now the terms have the same radicand so we can subtract. — — — ​ − 28​∣ a ∣​b2​√2ac ​ = 92​∣ a ∣​b2​√2ac ​ 120​∣ a ∣​b2​√2ac Try It #7 — — Subtract 3​√80x ​ − 4​√45x ​ .

To do this, we use the power rule of exponents. Consider the expression (x 2)3. The expression inside the parentheses is multiplied twice because it has an exponent of 2. Then the result is multiplied three times because the entire expression has an exponent of 3. 3 factors ​(x 2)​3​ = (x 2) · (x 2) · (x 2) 2 factors 3 factors 2 factors 2 factors = ​ ​ x · x ​ ​ · ​ ​ x · x ​ ​ · ​ ​ x · x ​ ​ =x·x·x·x·x·x = x 6 The exponent of the answer is the product of the exponents: ​(x 2)​3​ = x 2 ∙ 3 = x 6.

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