Let's go through each option and verify whether they are equivalent expressions.

1. $\frac{125m^5 p^5}{5m^5 p^{-3}} = 25m^{10} p^5$

   Simplify the left-hand side: $\frac{125}{5} = 25$ For $m^5$ in the numerator and $m^5$ in the denominator, they cancel out. So, we get: $\frac{m^5}{m^5} = 1$ For $p^5$ and $p^{-3}$, apply the rule $\frac{p^5}{p^{-3}} = p^{5 - (-3)} = p^{5 + 3} = p^8$.

   Therefore, the simplified expression is: $25 \times 1 \times p^8 = 25p^8$

   This is not equal to $25m^{10} p^5$. So, this option is incorrect.

2. $(5m^2 n^3)^0 = 1$

   Any non-zero number raised to the power of 0 is equal to 1.

   Therefore, this expression is correct.

3. $\sqrt[5]{x^2} = x^{\frac{2}{5}}$

   By the property of radicals, $\sqrt[n]{a^m} = a^{\frac{m}{n}}$. Here, $\sqrt[5]{x^2} = x^{\frac{2}{5}}$.

   This is a correct equivalent expression.

4. $8^{\frac{2}{3}} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4$

   Let's verify: $8^{\frac{2}{3}} = (2^3)^{\frac{2}{3}} = 2^{3 \times \frac{2}{3}} = 2^2 = 4$ The expression simplifies correctly to 4.

   This is a correct equivalent expression.

5. $\frac{8^3}{4^3} = \left(\frac{8}{4}\right)^3 = 2^3 = 8$

   Simplify $\frac{8^3}{4^3}$: $8 = 2^3, \quad 4 = 2^2$ Therefore, $\frac{8^3}{4^3} = \frac{(2^3)^3}{(2^2)^3} = \frac{2^9}{2^6} = 2^{9 - 6} = 2^3 = 8$

   This is a correct equivalent expression.

Correct Answers

• Options 2, 3, 4, and 5 are correct equivalent expressions.

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you simplify an expression with multiple variables and exponents?
2. What are the properties of exponents that help in simplifying expressions?
3. How can we convert radical expressions into exponent form?
4. What is the general rule for raising a power to another power?
5. How do we simplify a fraction with exponents in the numerator and denominator?

Tip: Always remember the basic exponent rules: product of powers, power of a power, and power of a product. They help simplify complex expressions easily.