Let’s analyze and solve the given expression step by step. Here's the equation extracted from the image:

$\frac{\left( -3^2 \right)^3 \cdot 27}{9^4 \cdot \left( \frac{1}{3} \right)}.$

Step 1: Simplify individual components

1. Simplify $(-3^2)^3$:

• First, calculate $-3^2$: $-3^2 = (-3) \cdot (-3) = 9.$
• Now, cube this result: $(9)^3 = 9 \cdot 9 \cdot 9 = 729.$ Thus, $(-3^2)^3 = 729$.

2. Simplify $27^{1/3}$:

The cube root of $27$ is: $27^{1/3} = 3.$

3. Simplify $9^4$:

$9^4 = (3^2)^4 = 3^{8}.$

4. Simplify the denominator with $\left( \frac{1}{3} \right)$:

When multiplying, $\frac{1}{3}$ remains as is.

Step 2: Rewrite the expression

Substitute the simplified components back into the equation: $\frac{729 \cdot 3}{3^8 \cdot \frac{1}{3}}.$

Step 3: Combine terms

Simplify the numerator:

$729 \cdot 3 = 2187.$

Simplify the denominator:

$3^8 \cdot \frac{1}{3} = 3^{8-1} = 3^7.$

The expression is now: $\frac{2187}{3^7}.$

Step 4: Simplify the fraction

Since $2187 = 3^7$, the fraction simplifies to: $\frac{3^7}{3^7} = 1.$

Final Answer:

$\boxed{1}$

Would you like a detailed breakdown of any particular step?

Related Questions:

1. How is the cube root of a number calculated?
2. Why does $9^4$ become $3^8$ when expressed with exponents of prime factors?
3. What are the steps to simplify expressions with negative bases and exponents?
4. How can we handle expressions involving fractions like $\frac{1}{3}$ more easily?
5. Why does $3^7 \div 3^7$ simplify to 1?

Tip:

When working with exponents, always express terms with the same base whenever possible—it simplifies calculations significantly!