Answer :
Let's go through Seth's steps to figure out where the mistake happened:
### Given Expression:
[tex]\[ 8 x^6 \sqrt{200 x^{13}} \div 2 x^5 \sqrt{32 x^7} \][/tex]
### Step 1:
Seth breaks down the square roots into prime factors:
[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 breakdown is correct. The square root components are broken down correctly.
### Step 2:
Seth combines constants and powers of [tex]\(x\)[/tex]:
[tex]\[ 8 \cdot 2 \cdot 5 \cdot x^6 \cdot x^6 \sqrt{2x} \div 2 \cdot 16 \cdot x^5 \cdot x^3 \sqrt{2x} \][/tex]
At this point, let’s look at what the simplified constants and [tex]\(x\)[/tex] powers should be. The mistake happens here because 32 was factored incorrectly:
- Instead of correctly simplifying the terms inside the square roots and constants correctly, Seth incorrectly combined them as [tex]\(2 \cdot 16\)[/tex].
### Correct Step 2 (without a mistake):
[tex]\[ 8 \cdot 2 \cdot 5 \cdot x^6 \cdot x^6 \sqrt{2x} \div 2 \cdot 4 \cdot x^5 \cdot x^3 \sqrt{2x} \][/tex]
### Step 3, 4, and 5:
These steps involve simplifying further, but because the error occurred in Step 2 with incorrect combination, it carried through the rest of the solution.
### Conclusion:
Seth's first mistake happened in Step 2, where he combined constants incorrectly by not correctly factoring the square terms. This led to an incorrect simplification in the subsequent steps. As a result, the final answer after correcting this mistake by using the correct factors will lead to the proper simplification of the expression.
### Given Expression:
[tex]\[ 8 x^6 \sqrt{200 x^{13}} \div 2 x^5 \sqrt{32 x^7} \][/tex]
### Step 1:
Seth breaks down the square roots into prime factors:
[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 breakdown is correct. The square root components are broken down correctly.
### Step 2:
Seth combines constants and powers of [tex]\(x\)[/tex]:
[tex]\[ 8 \cdot 2 \cdot 5 \cdot x^6 \cdot x^6 \sqrt{2x} \div 2 \cdot 16 \cdot x^5 \cdot x^3 \sqrt{2x} \][/tex]
At this point, let’s look at what the simplified constants and [tex]\(x\)[/tex] powers should be. The mistake happens here because 32 was factored incorrectly:
- Instead of correctly simplifying the terms inside the square roots and constants correctly, Seth incorrectly combined them as [tex]\(2 \cdot 16\)[/tex].
### Correct Step 2 (without a mistake):
[tex]\[ 8 \cdot 2 \cdot 5 \cdot x^6 \cdot x^6 \sqrt{2x} \div 2 \cdot 4 \cdot x^5 \cdot x^3 \sqrt{2x} \][/tex]
### Step 3, 4, and 5:
These steps involve simplifying further, but because the error occurred in Step 2 with incorrect combination, it carried through the rest of the solution.
### Conclusion:
Seth's first mistake happened in Step 2, where he combined constants incorrectly by not correctly factoring the square terms. This led to an incorrect simplification in the subsequent steps. As a result, the final answer after correcting this mistake by using the correct factors will lead to the proper simplification of the expression.
