Answer :
Let's go through the expression simplification process to identify where Seth made his first mistake:
Seth's original expression is:
[tex]\[
8x^6 \sqrt{200x^{13}} \div 2x^5 \sqrt{32x^7}
\][/tex]
Step 1: Prime factorization and breaking down under the square root
Seth rewrote:
- For [tex]\(\sqrt{200x^{13}}\)[/tex]:
- [tex]\(200\)[/tex] can be factored as [tex]\(4 \times 25 \times 2\)[/tex]
- [tex]\(x^{13}\)[/tex] can be split as [tex]\((x^6)^2 \times x\)[/tex]
So, [tex]\(\sqrt{200x^{13}}\)[/tex] turns into:
[tex]\[
\sqrt{4 \cdot 25 \cdot 2 \cdot (x^6)^2 \cdot x}
\][/tex]
- For [tex]\(\sqrt{32x^7}\)[/tex]:
- [tex]\(32\)[/tex] can be factored as [tex]\(16 \times 2\)[/tex]
- [tex]\(x^{7}\)[/tex] can be split as [tex]\((x^3)^2 \times x\)[/tex]
So, [tex]\(\sqrt{32x^7}\)[/tex] becomes:
[tex]\[
\sqrt{16 \cdot 2 \cdot (x^3)^2 \cdot x}
\][/tex]
Step 2: Simplifying the expression
- Taking square roots:
- [tex]\(\sqrt{4 \cdot 25} = 2 \cdot 5\)[/tex] and [tex]\(\sqrt{(x^6)^2} = x^6\)[/tex]
- [tex]\(\sqrt{16} = 4\)[/tex] and [tex]\(\sqrt{(x^3)^2} = x^3\)[/tex]
Seth's transformations:
[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]
He simplified the coefficients as [tex]\(2 \cdot 16\)[/tex], which seems off for this part, indicating a possible mistake. The correct multiplication of constants should be [tex]\(4\)[/tex], not [tex]\(16\)[/tex].
Identifying the Error
- It seems Seth mismanaged the constants within the square roots while simplifying from Step 1 to Step 2. Specifically, his mistake reveals itself in Step 1, where the constants and index divisions weren't handled accurately. Instead of interpreting that computation correctly, leading to errors in subsequent steps.
Therefore, the first mistake was made in Step 1, specifically on how the factors under the square roots were derived and simplified, setting up further errors.
Seth's original expression is:
[tex]\[
8x^6 \sqrt{200x^{13}} \div 2x^5 \sqrt{32x^7}
\][/tex]
Step 1: Prime factorization and breaking down under the square root
Seth rewrote:
- For [tex]\(\sqrt{200x^{13}}\)[/tex]:
- [tex]\(200\)[/tex] can be factored as [tex]\(4 \times 25 \times 2\)[/tex]
- [tex]\(x^{13}\)[/tex] can be split as [tex]\((x^6)^2 \times x\)[/tex]
So, [tex]\(\sqrt{200x^{13}}\)[/tex] turns into:
[tex]\[
\sqrt{4 \cdot 25 \cdot 2 \cdot (x^6)^2 \cdot x}
\][/tex]
- For [tex]\(\sqrt{32x^7}\)[/tex]:
- [tex]\(32\)[/tex] can be factored as [tex]\(16 \times 2\)[/tex]
- [tex]\(x^{7}\)[/tex] can be split as [tex]\((x^3)^2 \times x\)[/tex]
So, [tex]\(\sqrt{32x^7}\)[/tex] becomes:
[tex]\[
\sqrt{16 \cdot 2 \cdot (x^3)^2 \cdot x}
\][/tex]
Step 2: Simplifying the expression
- Taking square roots:
- [tex]\(\sqrt{4 \cdot 25} = 2 \cdot 5\)[/tex] and [tex]\(\sqrt{(x^6)^2} = x^6\)[/tex]
- [tex]\(\sqrt{16} = 4\)[/tex] and [tex]\(\sqrt{(x^3)^2} = x^3\)[/tex]
Seth's transformations:
[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]
He simplified the coefficients as [tex]\(2 \cdot 16\)[/tex], which seems off for this part, indicating a possible mistake. The correct multiplication of constants should be [tex]\(4\)[/tex], not [tex]\(16\)[/tex].
Identifying the Error
- It seems Seth mismanaged the constants within the square roots while simplifying from Step 1 to Step 2. Specifically, his mistake reveals itself in Step 1, where the constants and index divisions weren't handled accurately. Instead of interpreting that computation correctly, leading to errors in subsequent steps.
Therefore, the first mistake was made in Step 1, specifically on how the factors under the square roots were derived and simplified, setting up further errors.
