Let's analyze the pairs of expressions to determine which ones are equivalent.

1. $(4^3)^3$ and $4^3 \cdot 4^3$

• $(4^3)^3$ can be simplified using the power of a power rule: $(a^m)^n = a^{m \cdot n}$. $(4^3)^3 = 4^{3 \cdot 3} = 4^9$
• $4^3 \cdot 4^3$ can be simplified using the product of powers rule: $a^m \cdot a^n = a^{m+n}$. $4^3 \cdot 4^3 = 4^{3+3} = 4^6$ These two expressions are not equivalent because $4^9 \neq 4^6$.

2. $(3^4)^4$ and $3^8 \cdot 3^8$

• $(3^4)^4$ simplifies using the power of a power rule: $(3^4)^4 = 3^{4 \cdot 4} = 3^{16}$
• $3^8 \cdot 3^8$ simplifies using the product of powers rule: $3^8 \cdot 3^8 = 3^{8+8} = 3^{16}$ These two expressions are equivalent because both simplify to $3^{16}$.

3. $6^4 \cdot 3^4$ and $18^8$

• $6^4 \cdot 3^4$ cannot be combined directly because the bases are different. It remains as is: $6^4 \cdot 3^4$.
• $18^8$ cannot be rewritten in terms of $6^4$ or $3^4$. Thus, these expressions are not equivalent.

4. $4^3 \cdot 5^3$ and $20^3$

• $4^3 \cdot 5^3$ can be simplified by using the fact that $(a \cdot b)^n = a^n \cdot b^n$. $4^3 \cdot 5^3 = (4 \cdot 5)^3 = 20^3$ These two expressions are equivalent because both simplify to $20^3$.

5. $(4^3)^3$ and $4^3 \cdot 4^3 \cdot 4^3$

• $(4^3)^3$ simplifies to $4^9$ as shown earlier.
• $4^3 \cdot 4^3 \cdot 4^3$ simplifies using the product of powers rule: $4^3 \cdot 4^3 \cdot 4^3 = 4^{3+3+3} = 4^9$ These two expressions are equivalent because both simplify to $4^9$.

Final Answer:

The pairs of equivalent expressions are:

• $(3^4)^4$ and $3^8 \cdot 3^8$
• $4^3 \cdot 5^3$ and $20^3$
• $(4^3)^3$ and $4^3 \cdot 4^3 \cdot 4^3$

Would you like a detailed explanation of any of the rules used? Here are five related questions:

1. How do you apply the power of a product rule?
2. What is the power of a power rule in exponents?
3. How do you simplify expressions with the same base using multiplication?
4. Can you combine bases that are different when multiplying?
5. How would you simplify a negative exponent?

Tip: When multiplying expressions with the same base, add their exponents!