Let's solve the equation step by step:

$\left( 27 \right)^{5/2} \times \left( 9 \right)^{3/2} \div \left( 3 \right)^7 = \left( 3 \right)^x$

Step 1: Express everything in terms of powers of 3.

• $27 = 3^3$, so $(27)^{5/2} = \left( 3^3 \right)^{5/2} = 3^{15/2}$.
• $9 = 3^2$, so $(9)^{3/2} = \left( 3^2 \right)^{3/2} = 3^3$.

Thus, the equation becomes:

$3^{15/2} \times 3^3 \div 3^7 = 3^x$

Step 2: Use the laws of exponents.

• When multiplying powers of the same base, add the exponents: $3^{15/2} \times 3^3 = 3^{15/2 + 3} = 3^{15/2 + 6/2} = 3^{21/2}$.
• Now, divide powers of the same base by subtracting the exponents: $\frac{3^{21/2}}{3^7} = 3^{21/2 - 7} = 3^{21/2 - 14/2} = 3^{7/2}$.

Step 3: Set the exponents equal.

Since both sides have the base $3$, equate the exponents:

$\frac{7}{2} = x$

Thus, $x = \frac{7}{2}$.

Would you like further details or have any questions?

Here are 5 follow-up questions for deeper understanding:

1. What laws of exponents did we use in this problem?
2. How do you express 27 and 9 in terms of powers of 3?
3. What happens when you multiply two numbers with the same base but different exponents?
4. How do you divide powers with the same base?
5. What is the significance of rewriting terms in terms of a common base?

Tip: Whenever solving exponent problems, always try to express numbers with the same base to simplify calculations.