Let's solve the given equation step by step:

The equation is:

$\frac{3(3^2)^2}{3^5} = 3^8$

Step 1: Simplify the numerator.

In the numerator, we have $3(3^2)^2$. Let's break this down:

• $(3^2)^2 = 3^{2 \times 2} = 3^4$

Now, multiply this result by 3:

• $3 \times 3^4 = 3^1 \times 3^4 = 3^{1+4} = 3^5$

So, the numerator becomes $3^5$.

Step 2: Simplify the expression.

Now, the equation becomes:

$\frac{3^5}{3^5}$

Since both the numerator and denominator are the same, we have:

$\frac{3^5}{3^5} = 1$

Step 3: Compare with the right-hand side.

The right-hand side of the equation is $3^8$.

Thus, the equation simplifies to:

$1 \neq 3^8$

Conclusion:

The left-hand side simplifies to 1, while the right-hand side is $3^8$. Hence, the equation is false.

Would you like further details or have any questions?

Here are 5 related questions:

1. How does the exponentiation rule $(a^m)^n = a^{mn}$ apply to simplify expressions?
2. What are some strategies for simplifying exponents with the same base?
3. How do you handle equations that involve both division and multiplication of exponents?
4. Can you explain why dividing powers of the same base subtracts the exponents?
5. What are some common errors to avoid when simplifying exponent expressions?

Tip: Always remember that when dividing powers with the same base, you subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$.