for n and p > 0 is the expression (p^2n^1/2)^8 √p^5n^4 equivalent to p^18n^6√p? Justify your answer

Answer 1

You are missing a few clarifying parentheses, so I've made some assumptions...

(p² n^(1/2))⁸ = p^(2•8) n^((1/2)•8) = p¹⁶ n⁴ because when something to a power is raised to a power, the powers multiply and when a product is raised to a power, the power is distributed to each factor

the square root term can be written: (p⁵ n⁴)^(1/2) = p^5•(1/2) n^4•(1/2) = p² p^(1/2) n² because any radical can be written as a fractional power, and the same reasons as above, then the final step with p occurs because 5/2 = 2 + (1/2) and when you multiply the same variable to different powers, you add the powers.

Now combine the two parts:

(p¹⁶ n⁴)(p² p^(1/2) n²) = (p¹⁶ p²)(n⁴ n²) p^(1/2) = p⁽¹⁶⁺²⁾ n⁽⁴⁺²⁾ p^(1/2) = p¹⁸ n⁶ √p - because when multiplying the same variable with different powers, add the powers.

So yes they are the same.