Question

select the correct answer from each drop - down menu. review seth’s steps for rewriting and simplifying an expression. given $8x^{6}sqrt{200x^{13}}div2x^{5}sqrt{32x^{7}}$ step 1 $= 8x^{6}sqrt{4cdot25cdot2cdot(x^{6})^{2}cdot x}div2x^{5}sqrt{16cdot2cdot(x^{3})^{2}cdot x}$ step 2 $= 8cdot2cdot5cdot x^{6}cdot x^{6}sqrt{2x}div2cdot16cdot x^{5}cdot x^{3}sqrt{2x}$ step 3 $= 80x^{12}sqrt{2x}div32x^{8}sqrt{2x}$ step 4 $=\frac{80x^{12}sqrt{2x}}{32x^{8}sqrt{2x}}$ step 5 $=\frac{5}{2}x^{4}$ seth’s first mistake was made in $square$, where he $square$

Explanation:

Step1: Analyze Step1 factorization

$200x^{13}=4\cdot25\cdot2\cdot(x^6)^2\cdot x$, $32x^7=16\cdot2\cdot(x^3)^2\cdot x$ (correct)

Step2: Check Step2 simplification

$\sqrt{4\cdot25}=2\cdot5=10$, not $8\cdot2\cdot5$; $\sqrt{16}=4$, not $2\cdot16$.
Incorrect expansion: $8x^6\sqrt{4\cdot25\cdot2\cdot(x^6)^2\cdot x}=8x^6\cdot2\cdot5\cdot x^6\sqrt{2x}=80x^{12}\sqrt{2x}$ (coefficient error here: $\sqrt{4}=2$, $\sqrt{25}=5$, so $2\cdot5=10$, $8\cdot10=80$ is correct for first term, but second term: $2x^5\sqrt{16\cdot2\cdot(x^3)^2\cdot x}=2x^5\cdot4\cdot x^3\sqrt{2x}=8x^8\sqrt{2x}$, not $32x^8\sqrt{2x}$)

Step3: Identify first error step

The first miscalculation occurs in Step2, where he incorrectly simplified $\sqrt{16}$ as $16$ instead of $4$, leading to an inflated coefficient for the second term.

Answer:

Seth's first mistake was made in Step 2, where he incorrectly calculated the coefficient of the second radical term: he simplified $\sqrt{16}$ as $16$ (instead of $4$), resulting in $2 \cdot 16$ for the coefficient of the second expression, when it should be $2 \cdot 4$.