Textbook Question
Factor completely, or state that the polynomial is prime.
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 66
Evaluate each expression in Exercises 55–66, or indicate that the root is not a real number.
Verified step by step guidance1
Identify the expression to evaluate: the sixth root of \( \frac{1}{64} \), which can be written as \( \sqrt[6]{\frac{1}{64}} \).
Recall that the nth root of a number \( a \) can be expressed as \( a^{\frac{1}{n}} \). So, rewrite the expression as \( \left( \frac{1}{64} \right)^{\frac{1}{6}} \).
Express the denominator 64 as a power of a base number. Since \( 64 = 2^6 \), rewrite the expression as \( \left( \frac{1}{2^6} \right)^{\frac{1}{6}} \).
Apply the power of a power rule \( (a^m)^n = a^{m \times n} \) to simplify the expression: \( \left( 2^{-6} \right)^{\frac{1}{6}} = 2^{-6 \times \frac{1}{6}} \).
Simplify the exponent multiplication to get \( 2^{-1} \), which is the same as \( \frac{1}{2} \). This is the simplified form of the original expression.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Radical Expressions and Roots
Radical expressions involve roots such as square roots, cube roots, and higher-order roots. The notation ⁿ√a represents the nth root of a number a, which is the value that, when raised to the nth power, equals a. Understanding how to interpret and simplify these roots is essential for evaluating expressions like ⁶√(1/64).
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Radical Expressions with Fractions
Properties of Exponents
Roots can be expressed using fractional exponents, where ⁿ√a equals a^(1/n). This allows the use of exponent rules to simplify expressions. For example, ⁶√(1/64) can be rewritten as (1/64)^(1/6), facilitating easier calculation by breaking down the base and applying exponent rules.
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Real vs. Non-Real Roots
Not all roots yield real numbers; some may be complex or imaginary. For even roots, the radicand (the number inside the root) must be non-negative to have a real root. Since 1/64 is positive, its sixth root is real, but recognizing when roots are not real is important for correctly answering such questions.
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Related Practice
Textbook Question
Express the distance between the given numbers using absolute value. Then find the distance by evaluating the absolute value expression. 2 and 17
Textbook Question
In Exercises 57–64, factor using the formula for the sum or difference of two cubes. 8x^3+125
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Textbook Question
In Exercises 59–66, perform the indicated operations. Indicate the degree of the resulting polynomial. (3x4 y2+5x3 y−3y)−(2x4 y2−3x3 y−4y+6x)
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Textbook Question
Evaluate each expression 1251/3.
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Textbook Question
Write each number in decimal notation without the use of exponents.