- Simplify the inner fraction by applying the quotient rule for exponents: \(\frac{2ab}{a^{-6}b^2} = 2a^{1-(-6)}b^{1-2} = 2a^7b^{-1}\).
- Apply the power of a product rule by raising each component to the power of \(-3\): \(2^{-3}(a^7)^{-3}(b^{-1})^{-3}\).
- Compute the numerical value and product of exponents to get \(\frac{1}{8}a^{-21}b^3\).
- Rewrite with positive exponents to reach the final expression \(\boxed{\frac{b^3}{8 a^{21}}}\).
Explanation
- Simplifying the inside of the parentheses
To simplify the expression \(\left(\frac{2 a b}{a^{-6} b^2}\right)^{-3}\), we first focus on simplifying the term inside the parentheses. Using the quotient rule for exponents, \(\frac{x^m}{x^n} = x^{m-n}\), we can handle the variables \(a\) and \(b\) separately:
\[\frac{a^1}{a^{-6}} = a^{1 - (-6)} = a^{1+6} = a^7\]
\[\frac{b^1}{b^2} = b^{1-2} = b^{-1}\]
Substituting these back into the expression, the term inside the parentheses becomes:
\[\frac{2 a^7}{b^1} = 2 a^7 b^{-1}\]
- Applying the exponent rules
Now, we raise the simplified expression to the power of \(-3\):
\[\left(2 a^7 b^{-1}\right)^{-3}\]
Applying the power of a product rule \((xyz)^n = x^n y^n z^n\), we distribute the exponent \(-3\) to each factor:
\[2^{-3} \cdot (a^7)^{-3} \cdot (b^{-1})^{-3}\]
We calculate each part:
- \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\)
- \((a^7)^{-3} = a^{7 \times (-3)} = a^{-21}\)
- \((b^{-1})^{-3} = b^{-1 \times (-3)} = b^3\)
Combining these, we get:
\[\frac{1}{8} \cdot a^{-21} \cdot b^3 = \frac{b^3}{8 a^{21}}\]
- Final Conclusion
The result of the simplification is \(\frac{b^3}{8 a^{21}}\). Examining the options provided:
◯ \(\frac{b^3}{8 a^{10}}\)
◯ \(\frac{b^2}{8 a^{45}}\)
◯ \(\frac{a^6}{4 b}\)
◯ \(\frac{2 a^6}{b^5}\)
It appears that none of the provided choices match the calculated result \(\frac{b^3}{8 a^{21}}\). Based on the accurate step-by-step simplification of the given expression \(\left(\frac{2 a b}{a^{-6} b^2}\right)^{-3}\), the correct simplified form is \(\frac{b^3}{8 a^{21}}\).
Examples
In computer science and digital data storage, understanding how exponents work is crucial for calculating file sizes or memory capacities when operations involve scaling. For instance, when compressing a data stream that has a nested structure of parameters (similar to our expression inside the parentheses), you need to correctly calculate the total effect of scaling factors or power adjustments to ensure the final processed data is accurate. Mastering these laws of exponents helps in efficient resource management and algorithm design.