Simplify Powers of 8: Combining 8^(-3x), 8^(-3y), and 8^(2y+x)

To solve this problem, we need to simplify the expression $8^{-3x} \times 8^{-3y} \times 8^{2y+x}$ using exponent rules. Let's break it down step-by-step.

First, recognize that each term in the product has the same base, which is 8. The multiplication rule for exponents allows us to add the exponents when multiplying like bases. Our expression is:

$8^{-3x} \times 8^{-3y} \times 8^{2y+x}$

According to the rule $a^m \times a^n = a^{m+n}$, we add the exponents of each term:

$(-3x) + (-3y) + (2y + x)$

Combine the exponents inside the parentheses:

$-3x - 3y + 2y + x$

Now, group and simplify like terms:

• The terms with $x$: $-3x + x = -2x$
• The terms with $y$: $-3y + 2y = -y$

Combine these results:

The expression becomes $-2x - y$.

Therefore, the reduced form of the original expression is:

$8^{-2x-y}$