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Answer :
Sure! Let's go through Seth's steps for rewriting and simplifying the given expression step-by-step to identify where the first mistake was made:
Given Expression:
[tex]\[
8x^6 \sqrt{200x^{13}} \div 2x^5 \sqrt{32x^7}
\][/tex]
Step 1:
Seth rewrote it as:
[tex]\[
8x^6 \sqrt{4 \cdot 25 \cdot 2 \cdot (x^6)^2 \cdot x} \div 2x^5 \sqrt{16 \cdot 2 \cdot (x^3)^2 \cdot x}
\][/tex]
Here, Seth needed to correctly factorize the values under the square roots:
- [tex]\(200 = 4 \times 25 \times 2\)[/tex] is okay.
- And for [tex]\(x^{13}\)[/tex], writing it as [tex]\((x^6)^2 \cdot x\)[/tex] seems like a correct factorization, as [tex]\((x^6)^2 \cdot x = x^{12} \cdot x = x^{13}\)[/tex].
- [tex]\(32 = 16 \times 2\)[/tex] is fine.
- For [tex]\(x^7\)[/tex], he wrote it as [tex]\((x^3)^2 \cdot x\)[/tex], meaning [tex]\((x^3)^2 \cdot x = x^6 \cdot x = x^7\)[/tex].
Step 2:
Seth simplified and rewrote the expression as:
[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]
In this step, the coefficients and powers of [tex]\(x\)[/tex] were rearranged within the square roots correctly.
Step 3:
[tex]\[
80x^{12} \sqrt{2x} \div 32x^8 \sqrt{2x}
\][/tex]
He then moved to simplifying the expression further. Both parts have [tex]\(\sqrt{2x}\)[/tex], which can indeed be canceled out from numerator and denominator.
Step 4:
[tex]\[
\frac{80x^{12} \sqrt{2x}}{32x^8 \sqrt{2x}}
\][/tex]
This simplification correctly shows the division of coefficients and powers of [tex]\(x\)[/tex].
Step 5:
[tex]\[
\frac{5}{2} x^4
\][/tex]
After the division of the coefficients [tex]\(80/32\)[/tex] and simplifying the powers of [tex]\(x\)[/tex], this results in a finished simplified form.
Conclusion:
Seth's first mistake was in Step 1. Although the factorization did not seem incorrect here individually, close inspection suggests a problem with maintaining consistency across subsequent steps. The primary mistake may have been in not correctly handling the initial format transitioning to later steps with clear simplification, which might lead to misconceptions as observed in previous analysis.
Given Expression:
[tex]\[
8x^6 \sqrt{200x^{13}} \div 2x^5 \sqrt{32x^7}
\][/tex]
Step 1:
Seth rewrote it as:
[tex]\[
8x^6 \sqrt{4 \cdot 25 \cdot 2 \cdot (x^6)^2 \cdot x} \div 2x^5 \sqrt{16 \cdot 2 \cdot (x^3)^2 \cdot x}
\][/tex]
Here, Seth needed to correctly factorize the values under the square roots:
- [tex]\(200 = 4 \times 25 \times 2\)[/tex] is okay.
- And for [tex]\(x^{13}\)[/tex], writing it as [tex]\((x^6)^2 \cdot x\)[/tex] seems like a correct factorization, as [tex]\((x^6)^2 \cdot x = x^{12} \cdot x = x^{13}\)[/tex].
- [tex]\(32 = 16 \times 2\)[/tex] is fine.
- For [tex]\(x^7\)[/tex], he wrote it as [tex]\((x^3)^2 \cdot x\)[/tex], meaning [tex]\((x^3)^2 \cdot x = x^6 \cdot x = x^7\)[/tex].
Step 2:
Seth simplified and rewrote the expression as:
[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]
In this step, the coefficients and powers of [tex]\(x\)[/tex] were rearranged within the square roots correctly.
Step 3:
[tex]\[
80x^{12} \sqrt{2x} \div 32x^8 \sqrt{2x}
\][/tex]
He then moved to simplifying the expression further. Both parts have [tex]\(\sqrt{2x}\)[/tex], which can indeed be canceled out from numerator and denominator.
Step 4:
[tex]\[
\frac{80x^{12} \sqrt{2x}}{32x^8 \sqrt{2x}}
\][/tex]
This simplification correctly shows the division of coefficients and powers of [tex]\(x\)[/tex].
Step 5:
[tex]\[
\frac{5}{2} x^4
\][/tex]
After the division of the coefficients [tex]\(80/32\)[/tex] and simplifying the powers of [tex]\(x\)[/tex], this results in a finished simplified form.
Conclusion:
Seth's first mistake was in Step 1. Although the factorization did not seem incorrect here individually, close inspection suggests a problem with maintaining consistency across subsequent steps. The primary mistake may have been in not correctly handling the initial format transitioning to later steps with clear simplification, which might lead to misconceptions as observed in previous analysis.
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