Multiple Choice
Use the product rule to multiply the following.
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9. Roots, Radicals, and Complex Numbers
Simplifying Radical Expressions
Multiple Choice
Use the product rule to rewrite the term inside the radical as a product, then simplify.
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Verified step by step guidance1
Start with the expression inside the radical: \(\sqrt{72x^2}\). Use the product rule for square roots, which states that \(\sqrt{ab} = \sqrt{a} \times \sqrt{b}\), to rewrite this as \(\sqrt{72} \times \sqrt{x^2}\).
Recognize that \(\sqrt{x^2}\) simplifies to \(|x|\), but since we usually assume \(x\) is nonnegative in these contexts, this simplifies to \(x\).
Next, simplify \(\sqrt{72}\). Factor 72 into its prime factors or perfect squares: \(72 = 36 \times 2\), so \(\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2}\).
Since \(\sqrt{36} = 6\), rewrite \(\sqrt{72}\) as \(6\sqrt{2}\). Now the expression inside the radical becomes \(6\sqrt{2} \times x\).
Don't forget the negative sign outside the radical. So the entire expression simplifies to \(-6x\sqrt{2}\).
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