Question

select the correct answer from each drop-down menu.
review seth’s steps for rewriting and simplifying an expression.
given quad 8x^{6}sqrt{200x^{13}} div 2x^{5}sqrt{32x^{7}}
step 1 quad = 8x^{6}sqrt{4 cdot 25 cdot 2 cdot (x^{6})^{2} cdot x} div 2x^{5}sqrt{16 cdot 2 cdot (x^{3})^{2} cdot x}
step 2 quad = 8 cdot 2 cdot 5 cdot x^{6} cdot x^{6}sqrt{2x} div 2 cdot 16 cdot x^{5} cdot x^{3}sqrt{2x}
step 3 quad = 80x^{12}sqrt{2x} div 32x^{8}sqrt{2x}
step 4 quad = \frac{80x^{12}sqrt{2x}}{32x^{8}sqrt{2x}}
step 5 quad = \frac{5}{2}x^{4}
seth’s first mistake was made in quad \boxed{} , where he quad \boxed{}

Explanation:

Step1: Analyze Step 2

In Step 2, when simplifying the square roots, for the denominator's square root \(\sqrt{16\cdot2\cdot(x^3)^2\cdot x}\), the square root of 16 is 4, not 16. Also, for the variable part, \((x^3)^2 = x^6\), but Seth used \(x^3\) instead of \(x^6\) when taking the square root? Wait, no, let's re - check. Wait, the original denominator square root: \(\sqrt{32x^7}=\sqrt{16\cdot2\cdot x^6\cdot x}\) (since \(x^7=x^6\cdot x\)). So \(\sqrt{16\cdot2\cdot x^6\cdot x}=4x^3\sqrt{2x}\) (because \(\sqrt{x^6}=x^3\) as \(x^6=(x^3)^2\)). But in Step 2, Seth wrote \(2\cdot16\cdot x^5\cdot x^3\sqrt{2x}\) for the denominator part. Wait, the coefficient part: the original coefficient of the denominator is \(2x^5\), and when simplifying \(\sqrt{32x^7}\), the square root of 32 is \(\sqrt{16\cdot2}=4\sqrt{2}\), so the coefficient from the square root is 4, not 16. Also, the variable part: \(\sqrt{x^7}=\sqrt{x^6\cdot x}=x^3\sqrt{x}\), but in the expression, it's \(\sqrt{2x}\), so the variable part of the square root is \(x^3\), but the coefficient of the square root in the denominator's term: the original term is \(2x^5\sqrt{32x^7}\), so when simplifying \(\sqrt{32x^7}\), we have \(\sqrt{16\cdot2\cdot x^6\cdot x}=4x^3\sqrt{2x}\). So the denominator term should be \(2x^5\cdot4x^3\sqrt{2x}=8x^8\sqrt{2x}\), but Seth wrote \(2\cdot16\cdot x^5\cdot x^3\sqrt{2x}\). So in Step 2, when simplifying the square root of 32 (which is \(\sqrt{16\times2}\)), he took \(\sqrt{16}=16\) instead of 4, and also, the coefficient of the denominator's non - square - root part is 2, so multiplying 2 (from \(2x^5\)) with the square root's coefficient (which should be 4, not 16) is wrong. So the first mistake is in Step 2, where he incorrectly simplified the square root of 16 as 16 (instead of 4) and also made a mistake in the coefficient multiplication for the denominator.

Step2: Confirm the first mistake step

Looking at each step:

• Step 1: Factoring the radicands is correct. \(200x^{13}=4\times25\times2\times(x^6)^2\times x\) and \(32x^7 = 16\times2\times(x^3)^2\times x\), so Step 1 is correct.
• Step 2: When simplifying \(\sqrt{16\cdot2\cdot(x^3)^2\cdot x}\), \(\sqrt{16}=4\), but Seth used 16. Also, the term is \(2x^5\sqrt{32x^7}\), so after simplifying the square root, it's \(2x^5\times4x^3\sqrt{2x}=8x^8\sqrt{2x}\), but Seth wrote \(2\times16\times x^5\times x^3\sqrt{2x}\). So the first mistake is in Step 2.

Answer:

Seth's first mistake was made in Step 2, where he incorrectly simplified the square root of 16 as 16 (instead of 4) and made an error in the coefficient multiplication for the denominator term. (For the drop - down, the first box should be "Step 2", and the second box should be something like "incorrectly simplified \(\sqrt{16}\) as 16 (should be 4) and made a coefficient error in the denominator term")