Textbook Question
Concept Check Match each equation in Column I with its graph in Column II. I II 47. A. 48. B. 49. C. 50. (x + 3)² + (y + 2)² = 25 D.
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Ch. R - Algebra Review
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 1, Problem 51
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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Key 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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Introduction to Dot Product
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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03:40Quotients of Complex Numbers in Polar Form
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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6:20Example 6
Related Practice
Textbook Question
In the following exercises, (a) find the center-radius form of the equation of each circle described, and (b) graph it. See Examples 5 and 6. center (0, 0), radius 6
Textbook Question
Find each product. See Example 5. (4x² - 5y) (4x² + 5y)
Textbook Question
Determine whether each function is even, odd, or neither. See Example 5. ƒ(x) = 0.5x⁴ - 2x² + 6
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Textbook Question
Find each root. See Example 3. ∛0.125
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Textbook Question
Give (a) the additive inverse and (b) the absolute value of each number. -6⁄5