Textbook Question
Simplify each exponential expression in Exercises 23–64.
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Ch. P - Fundamental Concepts of Algebra
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 1, Problem 41
Use the product rule to simplify the expressions in Exercises 41 - 44. In exercises 43 - 44, assume that variables represent nonnegative real numbers. √300
Verified step by step guidance1
Recognize that the expression \( \sqrt{300} \) can be rewritten using exponents as \( 300^{\frac{1}{2}} \).
Factor 300 into its prime factors or into a product of a perfect square and another number. For example, \( 300 = 100 \times 3 \).
Use the property of radicals that \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \) to rewrite \( \sqrt{300} \) as \( \sqrt{100} \times \sqrt{3} \).
Simplify \( \sqrt{100} \) since 100 is a perfect square, which equals 10, so the expression becomes \( 10 \times \sqrt{3} \).
Express the simplified form as \( 10\sqrt{3} \), which is the simplified radical form of \( \sqrt{300} \).
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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 rule helps simplify expressions by breaking down numbers into factors whose roots are easier to compute.
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Expanding Radicals
Simplifying Square Roots
Simplifying square roots involves factoring the radicand into perfect squares and other factors, then taking the square root of the perfect squares outside the radical. For example, √300 can be simplified by factoring 300 into 100 * 3, giving 10√3.
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02:20Imaginary Roots with the Square Root Property
Nonnegative Real Numbers Assumption
Assuming variables represent nonnegative real numbers ensures that square roots are defined as principal (nonnegative) roots. This assumption allows the use of the product rule without considering complex numbers or negative roots.
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03:31Introduction to Complex Numbers
Related Practice
Textbook Question
In Exercises 15–58, find each product. (x+5)2
Textbook Question
In Exercises 39–48, factor the difference of two squares. x^2−144
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Textbook Question
In Exercises 33–68, add or subtract as indicated. 5/x + 3
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Textbook Question
In Exercises 15–58, find each product. (1−y5)(1+y5)
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Textbook Question
Factor the difference of two squares.