- Evaluate the statements using the power of a quotient rule \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\) and the power of a product rule \((ab)^n = a^n b^n\).
- The first statement, \(\left(\frac{k}{2}\right)^5 = \frac{k^5}{32}\), is true because \(2^5 = 32\).
- The second statement, \(\left(\frac{2b}{c}\right)^3 = \frac{8b^3}{c^3}\), is true because \(2^3 = 8\).
- The third and fourth statements are false because they incorrectly apply the rules.
- The correct statements are \(\boxed{\left(\frac{k}{2}\right)^5 = \frac{k^5}{32}, \left(\frac{2b}{c}\right)^3 = \frac{8b^3}{c^3}}\).
Explanation
- Analysis of Exponent Rules and Statements
To determine which statements are true, we utilize the fundamental laws of exponents: the power of a quotient rule, \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\) (where \(b
eq 0\)), and the power of a product rule, \((ab)^n = a^n b^n\). We will evaluate each statement individually using these rules.
- Evaluation of the First Statement
Evaluate the first statement: \(\left(\frac{k}{2}\right)^5 = \frac{k^5}{32}\). Applying the power of a quotient rule, we get:
\[\left(\frac{k}{2}\right)^5 = \frac{k^5}{2^5} = \frac{k^5}{32}\]
Since the left-hand side simplifies to the right-hand side, the first statement is true.
- Evaluation of the Second Statement
Evaluate the second statement: \(\left(\frac{2b}{c}\right)^3 = \frac{8b^3}{c^3}\). Applying the power of a quotient rule and then the power of a product rule, we have:
\[\left(\frac{2b}{c}\right)^3 = \frac{(2b)^3}{c^3} = \frac{2^3 b^3}{c^3} = \frac{8b^3}{c^3}\]
Since the left-hand side simplifies to the right-hand side, the second statement is true.
- Evaluation of the Third Statement
Evaluate the third statement: \(\left(\frac{w}{x}\right)^4 = \frac{1}{w^4 x^4}\). Applying the power of a quotient rule, we have:
\[\left(\frac{w}{x}\right)^4 = \frac{w^4}{x^4}\]
Comparing this to \(\frac{1}{w^4 x^4}\), we see they are not equivalent. Thus, the third statement is false.
- Evaluation of the Fourth Statement
Evaluate the fourth statement: \(\left(\frac{6}{e}\right)^2 = \frac{36}{e}\). Applying the power of a quotient rule, we have:
\[\left(\frac{6}{e}\right)^2 = \frac{6^2}{e^2} = \frac{36}{e^2}\]
Comparing this to \(\frac{36}{e}\), we see they are not equivalent. Thus, the fourth statement is false.
- Conclusion
In conclusion, the first and second statements are correct according to the laws of exponents. The final true statements are:
\(\left(\frac{k}{2}\right)^5 = \frac{k^5}{32}\) and \(\left(\frac{2b}{c}\right)^3 = \frac{8b^3}{c^3}\).
Examples
Understanding how to manipulate exponents correctly is essential in fields like physics and engineering, where complex formulas often involve ratios and powers. For example, when calculating the volume of a sphere \(\left(V = \frac{4}{3} \pi r^3\right)\) or scaling up a design where dimensions change by a certain factor, the power of a quotient rule ensures that scaling is applied to all components of the measurement correctly. Mastering these rules prevents errors in calculations involving scale factors, scientific notation, and algebraic modeling of real-world phenomena.