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Chapter 5: Problem 38

\(\left(\frac{w^{3}}{q^{3}}\right)^{2}\)

Short Answer

Expert verified
\(\frac{w^{6}}{q^{6}}\)

Step by step solution

01

- Apply the Power Rule

When raising a fraction to a power, apply the power to both the numerator and the denominator separately. Here, each term in the fraction \(\frac{w^{3}}{q^{3}}\) is raised to the power of 2.
02

- Simplify the Numerator

Raise the numerator to the power of 2: \( (w^{3})^{2} = w^{3 \cdot 2} = w^{6} \)
03

- Simplify the Denominator

Raise the denominator to the power of 2: \( (q^{3})^{2} = q^{3 \cdot 2} = q^{6} \)
04

- Combine the Results

After simplifying both the numerator and the denominator, the expression becomes: \(\frac{w^{6}}{q^{6}}\)

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Key Concepts

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

Power Rule
The Power Rule is a fundamental concept in algebra. It helps when dealing with exponents. Let's break down how it works. When you have an expression like \((w^3 / q^3)^2\), you must apply the power to each part separately. This is an essential step for solving such problems. This means you multiply the exponent inside by the exponent outside. For example, \((w^3)^2\) becomes \(w^{3*2} = w^6\). Similarly, \((q^3)^2\) becomes \(q^{3*2} = q^6\). Understanding this rule will help you simplify many expressions in algebra.
Simplifying Fractions
Simplifying fractions is another key concept. In our example, you end up with \(w^6 / q^6\). This fraction cannot be simplified further because the exponents are the same but the bases (w and q) are different. To simplify any fraction, look for common factors in the numerator and the denominator. If there are none, your fraction is already in its simplest form. Simplifying fractions can make your final answer clearer and easier to work with.
Numerator and Denominator
Understanding numerators and denominators is crucial for any fraction. The numerator is the top part, and the denominator is the bottom part. In our example, \(w^6\) is the numerator and \(q^6\) is the denominator. These terms tell you how many parts of a whole we are dealing with. Always simplify fractions by applying the same operations to both the numerator and the denominator. Remember these basics, and fractions will become much easier to manage.

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