- Simplify the term \((a^0 b^2)\) to \((1 \cdot b^2) = b^2\) since \(a^0 = 1\).
- Raise this to the power of 4 using the power rule \((x^m)^n = x^{mn}\), resulting in \((b^2)^4 = b^8\).
- Combine the results to get the final simplified expression \(\frac{6 a}{b^8}\).
- The correct choice is \(\boxed{\frac{6 a}{b^8}}\).
Explanation
- Simplifying the Base Expression
The expression provided is \(\frac{6 a}{(a^0 b^2)^4}\). To simplify this, we need to apply the fundamental laws of exponents.
First, recall that any non-zero number raised to the power of zero is 1, so \(a^0 = 1\).
Next, substitute this into the expression: \((a^0 b^2) = (1 \cdot b^2) = b^2\).
- Applying Power Laws and Finalizing the Expression
Now substitute the simplified base back into the full expression. The denominator becomes \((b^2)^4\). Using the power rule of exponents, which states \((x^m)^n = x^{mn}\), we have \((b^2)^4 = b^{2 \times 4} = b^8\).
Thus, the expression simplifies to \(\frac{6a}{b^8}\).
- Comparing with Options and Conclusion
Comparing our result \(\frac{6 a}{b^8}\) with the given options:
- \(\frac{6}{a^2 b^7}\)
- \(\frac{6}{a^2 b}\)
- \(\frac{6 a}{b}\)
- \(\frac{6 a}{b^8}\)
Our calculated expression perfectly matches the fourth option. Therefore, the equivalent expression is \(\frac{6 a}{b^8}\).
Examples
In data science and computational modeling, simplification is a common task when working with complex formulas involving physical constants or variable inputs. For example, when calculating the decay rate or a volume calculation in a physics model, reducing an expression like \(\frac{6 a}{(a^0 b^2)^4}\) to its simplest form \(\frac{6 a}{b^8}\) makes the final calculation much faster and less prone to errors in programming or spreadsheet software. Mastering these algebraic manipulations is a core skill for any field requiring quantitative analysis.