Let's analyze the given expressions and compare them.

We have three expressions to compare:

1. $3^3 \times 5^3$ on the left, compared to $15^3$ on the right.
2. $\frac{5^3 \times 3^3}{3^{15}}$ on the left, compared to $15^3$ on the right.
3. $\frac{5^3 \times 5^3}{15^{15}}$ on the left, compared to $15^{15}$ on the right.

Expression 1:

• Left side: $3^3 \times 5^3 = (3 \times 5)^3 = 15^3$
• Right side: $15^3$

Thus, $3^3 \times 5^3 = 15^3$, so we place the $=$ sign here.

Expression 2:

• Left side: $\frac{5^3 \times 3^3}{3^{15}} = \frac{(5 \times 3)^3}{3^{15}} = \frac{15^3}{3^{15}} = \frac{15^3}{3^5}$
• Right side: $15^3$

Comparing powers of 15 and 3, $\frac{15^3}{3^5}$ simplifies and is much smaller than $15^3$, so we place $<$ here.

Expression 3:

• Left side: $\frac{5^3 \times 5^3}{15^{15}} = \frac{5^6}{15^{15}}$
• Right side: $15^{15}$

Since $\frac{5^6}{15^{15}}$ is much smaller than $15^{15}$, we place $<$ here.

Final Answer:

• First comparison: $3^3 \times 5^3 = 15^3$
• Second comparison: $\frac{5^3 \times 3^3}{3^{15}} < 15^3$
• Third comparison: $\frac{5^3 \times 5^3}{15^{15}} < 15^{15}$

Would you like further clarification, or have any questions on this?

Here are five related questions to explore:

1. How do we simplify exponential expressions with the same base?
2. Why does $\frac{15^3}{3^{15}}$ simplify to a smaller value?
3. What happens when you multiply powers with the same base?
4. How do you compare powers of different bases?
5. Can the multiplication of powers be written as a single exponent?

Tip: When comparing expressions with powers, try simplifying the base and exponent first before making the comparison.