Amelia says that\sqrt{ }(4c^2)^5 simplifies to 32c^5Justify whether Amelia's answer is correct. If Amelia is not correct, explain what she should have done and find the correct answer.

Answer 1

Amelia's answer is not correct. To simplify the expression \sqrt{(4c^2)^5}, we need to first simplify what is inside the square root. The exponent on the expression 4c^2 is 5, so we can raise 4 to the power of 5 and c^2 to the power of 5, separately.

Starting with the base 4, we have 4^5, which is equal to 1024.

For the variable c^2, we have (c^2)^5, which is equal to c^(2*5) = c^10.

Now, putting it all together, we have \sqrt{(4c^2)^5} = \sqrt{1024c^10}.

Since 1024 is a perfect square, we can simplify the expression further by factoring out any perfect square factors from c^10. The largest perfect square factor of c^10 is c^10 = (c^2)^5, so we can rewrite c^10 as (c^2)^5.

Now, our expression becomes \sqrt{1024c^10} = \sqrt{1024(c^2)^5} = \sqrt{1024}(c^2)^{5/2} = 32c^5.

Therefore, the correct answer is 32c^5, which confirms that Amelia's answer is correct.