Textbook Question
Identify each set as finite or infinite. Then determine whether 10 is an element of the set. {1, 1/2, 1/4, 1/8, ....}
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0. Review of Algebra
Exponents
1:00 minutesProblem 14
Textbook Question
Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. √27
Verified step by step guidance1
Identify the expression to simplify, which is \( \sqrt{27} \). Recognize that \( \sqrt{27} \) can be rewritten using the property of square roots as \( \sqrt{27} = \sqrt{9 \times 3} \).
Apply the product rule for square roots, which states that \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \), to rewrite \( \sqrt{27} \) as \( \sqrt{9} \times \sqrt{3} \).
Simplify the square root of the perfect square: \( \sqrt{9} = 3 \). So the expression becomes \( 3 \times \sqrt{3} \).
Express the simplified form as \( 3\sqrt{3} \), which is the simplified radical form of \( \sqrt{27} \).
Confirm that the variables represent nonnegative real numbers, ensuring the square root expressions are valid and the simplification holds.
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Video duration:
1mKey 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 rule helps simplify expressions by breaking down complex radicals into simpler factors.
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Expanding Radicals
Simplifying Square Roots
Simplifying square roots involves expressing the radicand as a product of perfect squares and other factors, then taking the square root of the perfect squares outside the radical. For example, √27 = √(9 * 3) = 3√3.
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Nonnegative Real Numbers Assumption
Assuming variables represent nonnegative real numbers ensures that square roots are defined as principal (nonnegative) roots. This assumption avoids ambiguity in simplification and allows the use of the product rule without considering complex numbers.
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