Answer :
To solve the problem and identify Seth's first mistake, let's go through his steps carefully:
Given Expression:
[tex]\[ \frac{8 x^6 \sqrt{200 x^{13}}}{2 x^5 \sqrt{32 x^7}} \][/tex]
Step 1: Break down the components inside the square roots.
Seth rewrote:
1. [tex]\(\sqrt{200 x^{13}} = \sqrt{4 \cdot 25 \cdot 2 \cdot (x^6)^2 \cdot x}\)[/tex]
2. [tex]\(\sqrt{32 x^7} = \sqrt{16 \cdot 2 \cdot (x^3)^2 \cdot x}\)[/tex]
The breakdown looks correct.
Step 2: Simplify the factors.
Seth simplified to:
1. [tex]\(8 \cdot x^6 \cdot \sqrt{4 \cdot 25 \cdot 2 \cdot (x^6)^2 \cdot x} \rightarrow 8 \cdot 2 \cdot 5 \cdot x^6 \cdot x^6 \cdot \sqrt{2x}\)[/tex]
2. [tex]\(2 \cdot x^5 \cdot \sqrt{16 \cdot 2 \cdot (x^3)^2 \cdot x} \rightarrow 2 \cdot 4 \cdot x^5 \cdot x^3 \cdot \sqrt{2x}\)[/tex]
However, Seth wrote [tex]\(2 \cdot 16 \cdot x^5 \cdot x^3\)[/tex], which is incorrect. He should have used [tex]\(2 \cdot 4\)[/tex] instead of [tex]\(2 \cdot 16\)[/tex].
Step 3: Combine terms and simplify further.
This should be:
1. [tex]\(\frac{80 x^{12} \sqrt{2x}}{8 x^8 \sqrt{2x}}\)[/tex]
By evaluating honestly, Seth's step is showing calculations that led to combining [tex]\( 32 x^8 \)[/tex] instead of [tex]\( 8 x^8 \)[/tex].
Step 4: Write as a fraction:
This step looks correct:
[tex]\[ \frac{80 x^{12} \sqrt{2x}}{32 x^8 \sqrt{2x}} \][/tex]
Step 5: Final Simplification.
Simplify the fraction:
[tex]\[ \frac{80}{32} = \frac{5}{2} \quad \text{and} \quad \frac{x^{12}}{x^8} = x^4 \][/tex]
Resulting in:
[tex]\[ \frac{5}{2} x^4 \][/tex]
Conclusion:
Seth's first mistake was in Step 2 where he incorrectly multiplied by [tex]\(16\)[/tex] instead of using the correct breakdown which should have been [tex]\(2 \cdot 4\)[/tex]. This leads to errors in further steps.
Given Expression:
[tex]\[ \frac{8 x^6 \sqrt{200 x^{13}}}{2 x^5 \sqrt{32 x^7}} \][/tex]
Step 1: Break down the components inside the square roots.
Seth rewrote:
1. [tex]\(\sqrt{200 x^{13}} = \sqrt{4 \cdot 25 \cdot 2 \cdot (x^6)^2 \cdot x}\)[/tex]
2. [tex]\(\sqrt{32 x^7} = \sqrt{16 \cdot 2 \cdot (x^3)^2 \cdot x}\)[/tex]
The breakdown looks correct.
Step 2: Simplify the factors.
Seth simplified to:
1. [tex]\(8 \cdot x^6 \cdot \sqrt{4 \cdot 25 \cdot 2 \cdot (x^6)^2 \cdot x} \rightarrow 8 \cdot 2 \cdot 5 \cdot x^6 \cdot x^6 \cdot \sqrt{2x}\)[/tex]
2. [tex]\(2 \cdot x^5 \cdot \sqrt{16 \cdot 2 \cdot (x^3)^2 \cdot x} \rightarrow 2 \cdot 4 \cdot x^5 \cdot x^3 \cdot \sqrt{2x}\)[/tex]
However, Seth wrote [tex]\(2 \cdot 16 \cdot x^5 \cdot x^3\)[/tex], which is incorrect. He should have used [tex]\(2 \cdot 4\)[/tex] instead of [tex]\(2 \cdot 16\)[/tex].
Step 3: Combine terms and simplify further.
This should be:
1. [tex]\(\frac{80 x^{12} \sqrt{2x}}{8 x^8 \sqrt{2x}}\)[/tex]
By evaluating honestly, Seth's step is showing calculations that led to combining [tex]\( 32 x^8 \)[/tex] instead of [tex]\( 8 x^8 \)[/tex].
Step 4: Write as a fraction:
This step looks correct:
[tex]\[ \frac{80 x^{12} \sqrt{2x}}{32 x^8 \sqrt{2x}} \][/tex]
Step 5: Final Simplification.
Simplify the fraction:
[tex]\[ \frac{80}{32} = \frac{5}{2} \quad \text{and} \quad \frac{x^{12}}{x^8} = x^4 \][/tex]
Resulting in:
[tex]\[ \frac{5}{2} x^4 \][/tex]
Conclusion:
Seth's first mistake was in Step 2 where he incorrectly multiplied by [tex]\(16\)[/tex] instead of using the correct breakdown which should have been [tex]\(2 \cdot 4\)[/tex]. This leads to errors in further steps.
