Textbook Question
Write each number in decimal notation without the use of exponents.
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 73
Simplify the radical expressions in Exercises 67–74, if possible. ⁵√64x6/⁵√2x
Verified step by step guidance1
Rewrite the given expression as a single radical: \( \frac{\sqrt[5]{64x^6}}{\sqrt[5]{2x}} = \sqrt[5]{\frac{64x^6}{2x}} \).
Simplify the fraction inside the radical: \( \frac{64x^6}{2x} = 32x^5 \).
Substitute the simplified fraction back into the radical: \( \sqrt[5]{32x^5} \).
Break the radical into two parts: \( \sqrt[5]{32} \cdot \sqrt[5]{x^5} \).
Simplify each part: \( \sqrt[5]{32} = 2 \) (since \( 2^5 = 32 \)) and \( \sqrt[5]{x^5} = x \). Combine the results to get \( 2x \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Radical Expressions
Radical expressions involve roots, such as square roots or cube roots, represented by the radical symbol (√). In this context, we are dealing with fifth roots, denoted as ⁵√. Understanding how to manipulate these expressions, including simplifying them and applying properties of exponents, is crucial for solving problems involving radicals.
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Properties of Exponents
The properties of exponents are rules that govern how to handle expressions involving powers. Key properties include the product of powers (a^m * a^n = a^(m+n)) and the quotient of powers (a^m / a^n = a^(m-n)). These rules are essential for simplifying expressions with radicals, as they allow us to combine and reduce terms effectively.
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Simplifying Radicals
Simplifying radicals involves reducing a radical expression to its simplest form. This often includes factoring out perfect powers from under the radical and applying the properties of exponents. For example, when simplifying ⁵√(64x^6), recognizing that 64 is a perfect fifth power (2^6) and x^6 can be expressed as (x^5)(x) helps in reducing the expression.
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Related Practice
Textbook Question
In Exercises 65–92, factor completely, or state that the polynomial is prime. 6x2−6x−12
Textbook Question
Perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree.
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Textbook Question
In Exercises 67–82, find each product. (9x+7y)2
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Textbook Question
Express the distance between the given numbers using absolute value. Then find the distance by evaluating the absolute value expression. -26 and -3
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Textbook Question
In Exercises 69–82, simplify each complex rational expression. (1/x + 1/y)/(x+y)