Evaluate (5×6×8)^(-9): Negative Exponent Expression Problem

Insert the corresponding expression:

$\left(5\times6\times8\right)^{-9}=$

Let's solve the problem step-by-step to find the value of $(5 \times 6 \times 8)^{-9}$.

Step 1: Recognize that the expression $(5 \times 6 \times 8)^{-9}$ has a negative exponent. According to the negative exponent rule $a^{-b} = \frac{1}{a^b}$, we can write:

$(5 \times 6 \times 8)^{-9} = \frac{1}{(5 \times 6 \times 8)^9}$

Step 2: Next, apply the power of a product rule. This rule states that $(xyz)^n = x^n \times y^n \times z^n$. Therefore, apply this to $(5 \times 6 \times 8)^9$:

$(5 \times 6 \times 8)^9 = 5^9 \times 6^9 \times 8^9$

Step 3: Substitute back into the fraction obtained in Step 1:

$\frac{1}{(5 \times 6 \times 8)^9} = \frac{1}{5^9 \times 6^9 \times 8^9}$

This result is the fully simplified expression sought for the original problem.

Therefore, the expression $(5 \times 6 \times 8)^{-9}$ simplifies to $\frac{1}{5^9 \times 6^9 \times 8^9}$.