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

Question

Solve the following problem:

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

00:00 Simplify the following problem

00:05 Let's break down 9 into 3 squared

00:12 When there's a power over a power, the combined power is the multiplication of the powers

00:17 We'll apply this formula to our exercise, multiply between the powers

00:34 When multiplying powers with equal bases

00:38 The power of the result equals the sum of the powers

00:41 We'll apply this formula to our exercise, then add up the powers

00:54 According to the laws of powers, any number to the power of 0 equals 1

00:57 As long as the number is not 0

01:01 We'll apply this formula to our exercise

01:06 Any fraction/number raised to a negative power

01:09 We can invert the numerator and the denominator in order to obtain a positive power

01:12 We'll apply this formula to our exercise

01:15 That's the solution

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 $6^{-3}$ .

$6^{-3}$