Textbook Question
Factor completely, or state that the polynomial is prime. 64-x^2
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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 93
Simplify using properties of exponents.
Verified step by step guidance1
Identify the expression to simplify: \(\frac{20x^{\frac{1}{2}}}{5x^{\frac{1}{4}}}\).
Simplify the coefficients (numerical parts) by dividing 20 by 5: \(\frac{20}{5} = 4\).
Apply the quotient rule for exponents to the variable part: \(\frac{x^{\frac{1}{2}}}{x^{\frac{1}{4}}} = x^{\frac{1}{2} - \frac{1}{4}}\).
Subtract the exponents: \(\frac{1}{2} - \frac{1}{4} = \frac{2}{4} - \frac{1}{4} = \frac{1}{4}\), so the variable part becomes \(x^{\frac{1}{4}}\).
Combine the simplified coefficient and variable parts to write the simplified expression: \(4x^{\frac{1}{4}}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Properties of Exponents
Properties of exponents are rules that simplify expressions involving powers of the same base. Key properties include the quotient rule, which states that when dividing like bases, subtract the exponents (a^m / a^n = a^(m-n)). These rules help in simplifying expressions with fractional exponents.
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Fractional Exponents
Fractional exponents represent roots and powers simultaneously. For example, x^(1/2) means the square root of x, and x^(1/4) means the fourth root of x. Understanding how to manipulate fractional exponents is essential for simplifying expressions involving roots.
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Simplifying Rational Expressions
Simplifying rational expressions involves reducing fractions by factoring and canceling common terms. When variables with exponents appear in numerator and denominator, apply exponent rules to combine or reduce terms, making the expression simpler and easier to interpret.
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Related Practice
Textbook Question
Simplify using properties of exponents.
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Textbook Question
Perform the indicated computations. Write the answers in scientific notation. If necessary, round the decimal factor in your scientific notation answer to two decimal places.
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Textbook Question
In Exercises 93–102, factor and simplify each algebraic expression. x3/4−x1/4
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Textbook Question
Perform the indicated computations. Write the answers in scientific notation. If necessary, round the decimal factor in your scientific notation answer to two decimal places.
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Textbook Question
In Exercises 85–96, simplify each algebraic expression. 7−4[3−(4y−5)]
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