- Simplify the numerical coefficient: \(\frac{-4}{8} = -\frac{1}{2}\).
- Apply the quotient rule for powers of \(a\): \(\frac{a^{-2}}{a^{-6}} = a^{-2 - (-6)} = a^4\).
- Apply the quotient rule for powers of \(b\): \(\frac{b^4}{b^{-3}} = b^{4 - (-3)} = b^7\).
- Compare the result \(-\frac{1}{2} a^4 b^7\) to the given expression \(-\frac{1}{2} a^4 b^n\) to find that \(n = \boxed{7}\).
Explanation
- Analyzing the Expression
The given expression to simplify is \(\frac{-4 a^{-2} b^4}{8 a^{-6} b^{-3}}\) under the condition that \(a
eq 0\) and \(b
eq 0\). We are comparing the simplified form of this expression to the given form \(-\frac{1}{2} a^4 b^n\), where the task is to determine the unknown exponent \(n\). To simplify the original expression, we can separate it into three components: the numerical coefficients, the powers of \(a\), and the powers of \(b\).
- Simplifying the Expression Components
First, let us simplify the numerical coefficients by dividing \(-4\) by \(8\):
\[\frac{-4}{8} = -\frac{1}{2}\]
Next, we simplify the powers of \(a\) using the exponent rule \(\frac{a^m}{a^k} = a^{m-k}\):
\[\frac{a^{-2}}{a^{-6}} = a^{-2 - (-6)} = a^{-2 + 6} = a^4\]
Finally, we simplify the powers of \(b\) using the same exponent rule:
\[\frac{b^4}{b^{-3}} = b^{4 - (-3)} = b^{4 + 3} = b^7\]
- Determining the Value of n
By multiplying the simplified components together, we get:
\[-\frac{1}{2} \cdot a^4 \cdot b^7 = -\frac{1}{2} a^4 b^7\]
We were given that the expression simplifies to \(-\frac{1}{2} a^4 b^n\). By comparing our result \(-\frac{1}{2} a^4 b^7\) with the given form, we can clearly see that \(n = 7\). Therefore, the correct exponent \(n\) that Marina should use is \(7\).
Examples
Understanding exponent rules is essential for manipulating algebraic expressions in fields like computer science or engineering. For example, when calculating memory requirements or data storage sizes that involve powers of base units (like bits or bytes), simplifying expressions using exponent rules allows for quick and accurate calculations without needing to compute large numbers directly. If you have a expression related to file size scaling \(\frac{2^{10} \cdot x^5}{2^2 \cdot x^2}\), you can simplify it to \(2^{10-2} \cdot x^{5-2} = 2^8 \cdot x^3\) to easily understand how your data requirement grows.