Question

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review seths steps for rewriting and simplifying an expression.
given (\boldsymbol{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})
seths first mistake was made in (square), where he (square).

Explanation:

Step1: Analyze Step 1 factorization

Check radicand factorization:
$\sqrt{200x^{13}}=\sqrt{4\cdot25\cdot2\cdot(x^6)^2\cdot x}$, $\sqrt{32x^7}=\sqrt{16\cdot2\cdot(x^3)^2\cdot x}$
This step is correct.

Step2: Analyze Step 2 simplification

Simplify square roots:
$\sqrt{4\cdot25\cdot2\cdot(x^6)^2\cdot x}=2\cdot5\cdot x^6\sqrt{2x}=10x^6\sqrt{2x}$, so $8x^6\cdot10x^6\sqrt{2x}=80x^{12}\sqrt{2x}$
For the denominator: $\sqrt{16\cdot2\cdot(x^3)^2\cdot x}=4\cdot x^3\sqrt{2x}$, so $2x^5\cdot4x^3\sqrt{2x}=8x^8\sqrt{2x}$
Seth incorrectly used $16$ instead of $4$ here.

Step3: Identify first error location

The first mistake occurs in Step 2, as the simplification of $\sqrt{16\cdot2\cdot(x^3)^2\cdot x}$ was miscalculated (he used $16$ instead of $4$ when taking the square root of 16).

Answer:

Seth's first mistake was made in Step 2, where he incorrectly simplified $\sqrt{16}$ as $16$ instead of $4$, leading to an incorrect coefficient in the denominator term.