Textbook Question
Find each root. See Example 3. -∛512
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0. Review of College Algebra
Rationalizing Denominators
2:00 minutesProblem 51
Textbook Question
Use the product and quotient rules for radicals to rewrite each expression. See Example 4. √3 • √5
Verified step by step guidance1
Recall the product rule for radicals, which states that for any non-negative numbers \(a\) and \(b\), \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\).
Apply the product rule to the given expression \(\sqrt{3} \cdot \sqrt{5}\) by combining the radicals under a single square root: \(\sqrt{3 \cdot 5}\).
Multiply the numbers inside the radical: \(3 \cdot 5 = 15\), so the expression becomes \(\sqrt{15}\).
Check if the number inside the radical can be simplified further by factoring out perfect squares. Since 15 has no perfect square factors other than 1, it cannot be simplified further.
Therefore, the expression \(\sqrt{3} \cdot \sqrt{5}\) is rewritten as \(\sqrt{15}\) using the product rule for radicals.
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2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Product Rule for Radicals
The product rule for radicals states that the square root of a product equals the product of the square roots: √a • √b = √(a•b). This allows simplification by combining radicals under a single root, making expressions easier to handle.
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Quotient Rule for Radicals
The quotient rule for radicals states that the square root of a quotient equals the quotient of the square roots: √(a/b) = √a / √b. This rule helps in simplifying expressions involving division under radicals.
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Simplification of Radicals
Simplifying radicals involves rewriting expressions to their simplest form by factoring out perfect squares or combining terms using product and quotient rules. This process makes expressions clearer and easier to work with in further calculations.
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