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Ch. P - Fundamental Concepts of Algebra
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. P - Fundamental Concepts of AlgebraProblem 93
Chapter 1, Problem 93

Simplify using properties of exponents. 20x125x14\(\frac{20x^{\frac{1}{2}\)}}{5x^{\(\frac{1}{4}\)}}

Verified step by step guidance
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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

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. (6.1×10−8)(2×10−4)(6.1\(\times\)10^{-8})(2\(\times\)10^{-4})

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In Exercises 93–102, factor and simplify each algebraic expression. x3/4−x1/4

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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. (4.3×108)(6.2×104)(4.3×10^8)(6.2×10^4)

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In Exercises 85–96, simplify each algebraic expression. 7−4[3−(4y−5)]