Answered

Review Seth's steps for rewriting and simplifying an expression.



Given: $\quad 8 x^6 \sqrt{200 x^{13}} \div 2 x^5 \sqrt{32 x^7}$



Step $1: = 8 x^6 \sqrt{4 \cdot 25 \cdot 2 \cdot\left(x^6\right)^2 \cdot x} \div 2 x^5 \sqrt{16 \cdot 2 \cdot\left(x^3\right)^2 \cdot x}$



Step $2: = 8 \cdot 2 \cdot 5 \cdot x^6 \cdot x^6 \sqrt{2 x} \div 2 \cdot 16 \cdot x^5 \cdot x^3 \sqrt{2 x}$



Step $3: = 80 x^{12} \sqrt{2 x} \div 32 x^8 \sqrt{2 x}$



Step $4: = \frac{80 x^{12} \sqrt{2 x}}{32 x^8 \sqrt{2 x}}$



Step $5: = \frac{5}{2} x^4$



Seth's first mistake was made in [Choose an answer], where he [Choose an answer].

Verified Answer :

  • Analyze the original expression to find the correct simplified form: \(10 x^4\).
  • Review Seth's Step 1 and confirm it correctly decomposes the radicands.
  • Observe that in Step 2, Seth incorrectly calculates the coefficient for the second radical as \(16\) instead of \(4\), leading to a value of \(32\) instead of \(8\).
  • Conclude that the error occurs in Step 2, leading to an incorrect final answer of \(\frac{5}{2} x^4\) instead of \(10 x^4\).
  • Seth's first mistake was made in \(\boxed{\text{Step 2}}\), where he \(\boxed{\text{incorrectly simplified the square root of 32 by using 16 instead of 4}}\).

Explanation

  1. Analyzing the original expression

To find the mistake, we analyze the expression: \[8 x^6 \sqrt{200 x^{13}} \div 2 x^5 \sqrt{32 x^7}.\]
First, simplify the square roots:
\[\sqrt{200 x^{13}} = \sqrt{100 \cdot 2 \cdot x^{12} \cdot x} = 10 x^6 \sqrt{2x}\]
\[\sqrt{32 x^7} = \sqrt{16 \cdot 2 \cdot x^6 \cdot x} = 4 x^3 \sqrt{2x}\]
Now, evaluate the full expression:
\[8 x^6 \cdot (10 x^6 \sqrt{2x}) \div 2 x^5 \cdot (4 x^3 \sqrt{2x})\]
\[= 80 x^{12} \sqrt{2x} \div 8 x^8 \sqrt{2x}\]
\[= \frac{80 x^{12} \sqrt{2x}}{8 x^8 \sqrt{2x}} = 10 x^4.\]

  1. Comparing steps and identifying the mistake

Now compare this with Seth's steps:
Step 1: \(8 x^6 \sqrt{4 \cdot 25 \cdot 2 \cdot (x^6)^2 \cdot x} \div 2 x^5 \sqrt{16 \cdot 2 \cdot (x^3)^2 \cdot x}\).
This matches the correct expansion of the radicals: \(\sqrt{200 x^{13}} = \sqrt{4 \cdot 25 \cdot 2 \cdot (x^6)^2 \cdot x}\) and \(\sqrt{32 x^7} = \sqrt{16 \cdot 2 \cdot (x^3)^2 \cdot x}\). This step is correct.

Step 2: \(8 \cdot 2 \cdot 5 \cdot x^6 \cdot x^6 \sqrt{2 x} \div 2 \cdot 16 \cdot x^5 \cdot x^3 \sqrt{2 x}\).
Let's simplify the first part: \(8 \cdot (2 \cdot 5 \cdot x^6 \cdot x^6 \sqrt{2 x}) = 80 x^{12} \sqrt{2x}\). This part is correct.
However, look at the second part: \(2 \cdot 16 \cdot x^5 \cdot x^3 \sqrt{2 x} = 32 x^8 \sqrt{2 x}\).
Comparing this to our simplified term \(2 x^5 \sqrt{32 x^7} = 2 x^5 \cdot (4 x^3 \sqrt{2 x}) = 8 x^8 \sqrt{2 x}\), we see that Seth multiplied by 16 instead of 4. This is the mistake.

  1. Conclusion

In Step 2, Seth incorrectly simplified \(\sqrt{16 \cdot 2 \cdot (x^3)^2 \cdot x}\) by using 16 instead of 4 as the coefficient of the square root. Specifically, \(\sqrt{16 \cdot 2 \cdot (x^3)^2 \cdot x} = 4 x^3 \sqrt{2 x}\), but he treated it as \(16 x^3 \sqrt{2 x}\) (implicitly, by writing \(2 \cdot 16 \cdot x^5 \cdot x^3 \sqrt{2 x}\)). Consequently, all subsequent steps were affected by this error, leading to an incorrect final result. Therefore, Seth's first mistake was made in Step 2, where he incorrectly simplified the expression by multiplying by the wrong constant when extracting the square root of 32.

Examples

Mathematical verification is a crucial skill in fields such as software engineering and data analysis. Just as Seth made an error in a step-by-step simplification, a developer might introduce a bug in an algorithm by misapplying a constant in a loop. By carefully reviewing each line of code or logic, errors can be identified and corrected, ensuring that the final output is accurate. This process of systematic verification, often called debugging or code review, is essential for building reliable systems.

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