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.
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.