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Chapter 8: Problem 58

Simplify. \(\left(\frac{2 x w^{-3}}{3 x^{-4} w}\right)^{-2}\)

Short Answer

Expert verified
\(\frac{9 w^8}{4 x^{10}}\)

Step by step solution

01

Simplify the expression inside the parentheses

We start with the expression: ewline \(\frac{2 x w^{-3}}{3 x^{-4} w}\). ewline First, simplify the numerator and the denominator separately. In the numerator, we have 2, x, and \(w^{-3}\); in the denominator, we have 3, \(x^{-4}\), and w. Combine like terms using the properties of exponents.
02

Apply properties of exponents

Combine the x terms: ewline \(x \div x^{-4} = x^{1 - (-4)} = x^{1 + 4} = x^5\). ewline Combine the w terms: ewline \(w^{-3} \div w = w^{-3 - 1} = w^{-4}\). ewline The expression simplifies to: ewline \(\frac{2 x^5 w^{-4}}{3}\).
03

Apply the power of -2 to the simplified expression

Raise the simplified expression to the power of -2: ewline \(\frac{2 x^5 w^{-4}}{3}\)^{-2}. ewline Apply the exponent to each component inside the fraction separately: ewline \((2)^{-2}\), \((x^5)^{-2}\), \((w^{-4})^{-2}\), and \((3)^{-2}\).
04

Simplify each component

Simplify each component with the power of -2: ewline \(2^{-2} = \frac{1}{2^2} = \frac{1}{4}\), ewline \((x^5)^{-2} = x^{5 \times -2} = x^{-10}\), ewline \((w^{-4})^{-2} = w^{-4 \times -2} = w^8\), ewline \(3^{-2} = \frac{1}{3^2} = \frac{1}{9}\).
05

Combine all simplified components

Combine all simplified components back into the fraction: ewline \(\frac{\frac{1}{4} x^{-10} w^8}{\frac{1}{9}}\). ewline This simplifies to: ewline \(\frac{1}{4} \times \frac{9}{1} x^{-10} w^8 = \frac{9}{4} x^{-10} w^8\).
06

Reformat the expression to ensure positive exponents

Since \(x^{-10}\) has a negative exponent, reformat it so it has a positive exponent: ewline \(\frac{9 w^8}{4 x^{10}}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

properties of exponents
Simplifying fractions is crucial for making expressions manageable: Start with the numerator and the denominator separately, simplify each part, and then recombine: a.per the example, we combined like terms by raising the elements to a negative power:
  • This formed a simplified fraction. Next, each part of the simplified fraction is raised to the required power:
    • Repeat the conversion steps as needed. Always double-check your work to avoid common mistakes. ewlineUXI:Both x and we're neatly combined to form the simplified ...

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