Answer :
To identify Seth's first mistake in simplifying the expression, let's carefully go through each of his steps.
The given expression was:
[tex]\[ 8 x^6 \sqrt{200 x^{13}} \div 2 x^5 \sqrt{32 x^7} \][/tex]
Let's examine Seth's steps:
Step 1: Rewriting the square roots as:
[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]
This step correctly breaks down the square roots into prime factors, though it's somewhat unnecessarily complex.
Step 2: Simplifies to:
[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]
This step simplifies the square roots inside, but it looks like Seth made a mistake in the simplification here.
Step 3: Simplifies to:
[tex]\[ 80 x^{12} \sqrt{2 x} \div 32 x^8 \sqrt{2 x} \][/tex]
Here, Seth cancels out common terms, but observe that there is an inconsistency because later steps show cancellation and final values that don't result in what's expected from here.
Step 4: Further simplifies incorrectly:
[tex]\[ \frac{80 x^{11} \sqrt{2 \pi}}{32 x^3 \sqrt{2 x}} \][/tex]
Pi ([tex]\(\pi\)[/tex]) has unexpectedly appeared in the square root, indicating an error in copying or arithmetic before this.
Step 5: Final result:
[tex]\[ \frac{5}{2} x^4 \][/tex]
The first mistake happened in Step 4. The mistake was the introduction of [tex]\(\pi\)[/tex] in [tex]\(\sqrt{2 \pi}\)[/tex], which does not appear in any previous steps and is completely incorrect.
In summary, Seth's first error occurred in Step 4, where he incorrectly introduced [tex]\(\pi\)[/tex] in the square root.
The given expression was:
[tex]\[ 8 x^6 \sqrt{200 x^{13}} \div 2 x^5 \sqrt{32 x^7} \][/tex]
Let's examine Seth's steps:
Step 1: Rewriting the square roots as:
[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]
This step correctly breaks down the square roots into prime factors, though it's somewhat unnecessarily complex.
Step 2: Simplifies to:
[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]
This step simplifies the square roots inside, but it looks like Seth made a mistake in the simplification here.
Step 3: Simplifies to:
[tex]\[ 80 x^{12} \sqrt{2 x} \div 32 x^8 \sqrt{2 x} \][/tex]
Here, Seth cancels out common terms, but observe that there is an inconsistency because later steps show cancellation and final values that don't result in what's expected from here.
Step 4: Further simplifies incorrectly:
[tex]\[ \frac{80 x^{11} \sqrt{2 \pi}}{32 x^3 \sqrt{2 x}} \][/tex]
Pi ([tex]\(\pi\)[/tex]) has unexpectedly appeared in the square root, indicating an error in copying or arithmetic before this.
Step 5: Final result:
[tex]\[ \frac{5}{2} x^4 \][/tex]
The first mistake happened in Step 4. The mistake was the introduction of [tex]\(\pi\)[/tex] in [tex]\(\sqrt{2 \pi}\)[/tex], which does not appear in any previous steps and is completely incorrect.
In summary, Seth's first error occurred in Step 4, where he incorrectly introduced [tex]\(\pi\)[/tex] in the square root.
