Simplify the Expression: 8^4 ÷ 8^9 Using Laws of Exponents

To simplify the expression $\frac{8^4}{8^9}$, we apply the rule of exponents for division:

• The quotient rule for exponents is $\frac{a^m}{a^n} = a^{m-n}$.

Since both the numerator and the denominator have the same base (8), we can apply this rule directly:

$\frac{8^4}{8^9} = 8^{4-9}$

Thus, the resulting expression is $8^{-5}$.

Reviewing the choices given:

• Choice 1: $8^{9-4}$ which equals $8^5$, is incorrectly stating the subtraction order.
• Choice 2: $8^{\frac{4}{9}}$ is incorrect, as it represents a different operation (taking the root) rather than division of exponents.
• Choice 3: $8^{4-9}$ is correct, as it correctly applies the quotient rule for exponents.
• Choice 4: $8^{4\times9}$ suggests multiplication of exponents, not applicable here.

Therefore, the correct answer is Choice 3: $8^{4-9}$, which simplifies to $8^{-5}$.