The confusion here lies in how negative exponents work. Let’s break it down step by step.

Negative Exponent Rule:

The rule is that $a^{-n} = \frac{1}{a^n}$, which means a negative exponent in the denominator turns into a positive exponent when moved to the numerator. Conversely, a negative exponent in the numerator turns into a positive exponent when moved to the denominator.

Now let's analyze both expressions.

Expression 1:

$\frac{5}{a^{-8}b^{-1}} = 5 \times a^8 \times b$

• Start with the original form: $\frac{5}{a^{-8}b^{-1}}$
• By the negative exponent rule, $a^{-8}$ in the denominator moves to the numerator as $a^8$, and $b^{-1}$ moves to the numerator as $b^1$.
• The result becomes $5 \times a^8 \times b$.

Expression 2:

$5 \times a^{-8} \times b^{-1}$

• In this case, the negative exponents are already in the numerator, so they don't need to be moved.
• The result is $5 \times a^{-8} \times b^{-1}$, which stays as it is.

Key Insight:

The difference between these two forms is in whether the negative exponents are in the numerator or the denominator. In the first case, the negative exponents in the denominator become positive in the numerator. In the second case, the negative exponents are already in the numerator, so they remain negative.

So, to answer your question: $\frac{5}{a^{-8}b^{-1}} = 5 \times a^8 \times b$ and not $5 \times a^8 \times b$ because in the second case, the exponents were initially negative in the numerator.

Would you like further details, or do you have any questions?

Here are some related questions to explore:

1. What happens if there’s a negative exponent in both the numerator and the denominator?
2. How would you simplify a fraction like $\frac{a^{-5}}{b^{-3}}$?
3. What’s the effect of a zero exponent in expressions like $x^0$?
4. How do you simplify expressions with fractional exponents?
5. How do you handle negative exponents when variables are inside parentheses?

Tip: Always remember that moving terms with negative exponents between the numerator and denominator changes the sign of the exponent!