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Review Seth's steps for rewriting and simplifying an expression:

Given: [tex]\[8 x^6 \sqrt{200 x^{13}} \div 2 x^5 \sqrt{32 x^7}\][/tex]

Step 1: [tex]\[8 x^6 \sqrt{4 \cdot 25 \cdot 2 \cdot (x^6)^2 \cdot x} \div 2 x^5 \sqrt{16 \cdot 2 \cdot (x^3)^2 \cdot x}\][/tex]

Step 2: [tex]\[8 \cdot 2 \cdot 5 \cdot x^6 \cdot x^6 \sqrt{2 x} \div 2 \cdot 16 \cdot x^5 \cdot x^3 \sqrt{2 x}\][/tex]

Step 3: [tex]\[80 x^{12} \sqrt{2 x} \div 32 x^8 \sqrt{2 x}\][/tex]

Step 4: [tex]\[\frac{80 x^{12} \sqrt{2 x}}{32 x^8 \sqrt{2 x}}\][/tex]

Step 5: [tex]\[\frac{5}{2} x^4\][/tex]

Seth's first mistake was made in Step 2, where he [tex]$\square$[/tex].

We start by examining the expression:

[tex]$$
8 x^6 \sqrt{200 x^{13}} \div 2 x^5 \sqrt{32 x^7}.
$$[/tex]

In the process of simplifying, one key step is to factor the quantities inside the square roots to identify perfect squares.

For example, notice that in the denominator the radicand is

[tex]$$
32 x^7 = 16 \cdot 2 \cdot (x^3)^2 \cdot x.
$$[/tex]

When we take the square root, we correctly separate it as follows:

[tex]$$
\sqrt{32 x^7} = \sqrt{16} \cdot \sqrt{2} \cdot \sqrt{(x^3)^2} \cdot \sqrt{x}.
$$[/tex]

Here, since

[tex]$$
\sqrt{16} = 4 \quad \text{and} \quad \sqrt{(x^3)^2} = x^3,
$$[/tex]

the correctly simplified form becomes

[tex]$$
\sqrt{32 x^7} = 4 x^3 \sqrt{2x}.
$$[/tex]

However, in Seth’s step 2, he incorrectly wrote the factor from the square root of [tex]$16$[/tex] as [tex]$16$[/tex] (i.e., he used [tex]$16$[/tex] in place of the correct value [tex]$4$[/tex]). This error caused him to introduce an extra factor in the denominator, leading to a wrong overall multiplication factor.

Thus, Seth's first mistake is that he replaced [tex]$\sqrt{16}$[/tex] with [tex]$16$[/tex].