- Simplify the coefficient \(\frac{-8}{10}\) to \(-\frac{4}{5}\).
- Use the exponent quotient rule \(\frac{x^m}{x^n} = x^{m-n}\) to simplify \(a^{8}/a^{-4}\) as \(a^{12}\) and \(b^{-2}/b^{-10}\) as \(b^8\).
- Multiply the simplified components to get the result.
- The final answer is \(\boxed{-\frac{4 a^{12} b^8}{5}}\).
Explanation
- Analysis of the Expression
To simplify the expression \(\frac{-8 a^8 b^{-2}}{10 a^{-4} b^{-10}}\), where \(a
eq 0\) and \(b
eq 0\), we analyze the coefficients and the variable parts separately. The expression can be decomposed into a product of three distinct parts: the constant coefficients, the powers of \(a\), and the powers of \(b\). This allows us to apply the laws of exponents and arithmetic reduction systematically to each component.
- Simplifying Coefficients and Variables
First, we simplify the constant coefficient by reducing the fraction \(\frac{-8}{10}\). By dividing both the numerator and the denominator by their greatest common divisor, which is 2, we obtain:
\[\frac{-8}{10} = \frac{-8 \div 2}{10 \div 2} = -\frac{4}{5}.\]
Next, we simplify the powers of \(a\) using the quotient rule for exponents, \(\frac{x^m}{x^n} = x^{m-n}\):
\[\frac{a^8}{a^{-4}} = a^{8 - (-4)} = a^{8+4} = a^{12}.\]
Similarly, we simplify the powers of \(b\):
\[\frac{b^{-2}}{b^{-10}} = b^{-2 - (-10)} = b^{-2+10} = b^8.\]
- Conclusion and Final Form
Now, we combine the simplified components together:
\[\left(-\frac{4}{5}\right) \cdot a^{12} \cdot b^8 = -\frac{4 a^{12} b^8}{5}.\]
Comparing this result to the given options, we identify that it matches the first option. Therefore, the simplified form of the given expression is \(-\frac{4 a^{12} b^8}{5}\).
Examples
Simplifying expressions with exponents is a fundamental skill in algebra, used frequently in physics and engineering. For instance, if you are calculating the force between two charged particles using Coulomb's Law, the formula involves various powers of distances and constants. Properly reducing these algebraic expressions allows you to obtain a clear, concise final value that is much easier to work with when performing further calculations or interpreting scientific results.