Simplify the Expression: (9² × 3⁻⁴) ÷ 6³ Step by Step

Solve the following problem:

$\frac{9^2\cdot3^{-4}}{6^3}=\text{?}$

The problem requires simplification $\frac{9^2 \cdot 3^{-4}}{6^3}$ using exponent rules. Here’s a step-by-step guide to solving it:

• Step 1: Convert each term to powers of a common base.

  Notice that $9$ is $3^2$ and $6$ is $2 \times 3$. Hence:

  $9^2 = (3^2)^2 = 3^{4}$

  Therefore, the expression becomes $\frac{3^4 \cdot 3^{-4}}{(2 \cdot 3)^3}$.

• Step 2: Simplify the numerator.

  Using the exponent multiplication rule: $3^4 \cdot 3^{-4} = 3^{4 + (-4)} = 3^0 = 1$.

• Step 3: Expand the denominator.

  Calculate $(2 \cdot 3)^3$ by applying the distributive property: $(2 \cdot 3)^3 = 2^3 \cdot 3^3$.

• Step 4: Simplify the expression.

  After simplifying, the entire expression is $\frac{1}{2^3 \cdot 3^3}$.

  This simplifies further to $\frac{1}{6^3} = 6^{-3}$, because $2^3 \cdot 3^3 = (2 \cdot 3)^3 = 6^3$.

Therefore, the solution to the problem is $6^{-3}$.