Final answer:
The expressions (3^4)^4 and 3^8 * 3^8, as well as 4^3 * 5^3 and 20^3, are equivalent based on the rules of exponents. Others are not equivalent as they do not adhere to exponent multiplication and addition rules.
Step-by-step explanation:
The question requires an understanding of how to work with exponents and exponential expressions in mathematics. We will identify which pairs of expressions are equivalent by applying the rules of exponents.
a) (4^3)^3 and 4^3 * 4^3
When raising a power to another power, you multiply the exponents: (4^3)^3 = 4^(3*3) = 4^9. However, when you multiply two expressions with the same base, you add the exponents: 4^3 * 4^3 = 4^(3+3) = 4^6. Hence, these expressions are not equivalent.
b) (3^4)^4 and 3^8 * 3^8
Similarly, for expression (3^4)^4, you multiply the exponents: (3^4)^4 = 3^(4*4) = 3^16. For the second part, adding the exponents gives us 3^8 * 3^8 = 3^(8+8) = 3^16. Therefore, these expressions are equivalent.
c) 6^4 * 3^4 and 18^8
For this pair, we can use the property of exponents to combine the bases since they are being raised to the same power: 6^4 * 3^4 = (6*3)^4 = 18^4, which is not equivalent to 18^8.
d) 4^3 * 5^3 and 20^3
Using the property that allows us to combine bases being raised to the same power, we get 4^3 * 5^3 = (4*5)^3 = 20^3. Hence, these expressions are equivalent.