Answer :
The first mistake was in Step 2, where Seth incorrectly simplified the second term when taking the square root. The correct simplification should have resulted in $8x^8\sqrt{2x}$, but Seth wrote $32x^8\sqrt{2x}$. Therefore, Seth's first mistake was made in Step 2, where he incorrectly simplified the second term.
### Explanation
1. Problem Analysis
We are asked to identify the first mistake in Seth's simplification of the expression $8 x^6
\sqrt{200 x^{13}}
\div 2 x^5
\sqrt{32 x^7}$. We will analyze each step to find the error.
2. Step 1 Analysis
Step 1: $8 x^6
\sqrt{200 x^{13}}
\div 2 x^5
\sqrt{32 x^7} = 8 x^6
\sqrt{4 \cdot 25 \cdot 2 \cdot(x^6)^2 \cdot x}
\div 2 x^5
\sqrt{16 \cdot 2 \cdot(x^3)^2 \cdot x}$.
This step correctly factors the numbers and the powers of x inside the square roots.
3. Step 2 Analysis
Step 2: $8 x^6
\sqrt{4 \cdot 25 \cdot 2 \cdot(x^6)^2 \cdot x} = 8 x^6
\sqrt{100 \cdot 2 \cdot x^{12} \cdot x} = 8 x^6 \cdot 10 x^6
\sqrt{2x} = 80 x^{12}
\sqrt{2x}$. This part is correct.
Also, $2 x^5
\sqrt{16 \cdot 2 \cdot(x^3)^2 \cdot x} = 2 x^5
\sqrt{16 \cdot 2 \cdot x^6 \cdot x} = 2 x^5 \cdot 4 x^3
\sqrt{2x} = 8 x^8
\sqrt{2x}$.
Seth wrote $2 \cdot 16 \cdot x^5 \cdot x^3
\sqrt{2 x}$ which is incorrect. The 16 should be a 4, and the product should be $8x^8$ and not $32x^8$. Therefore, the first mistake was in Step 2, where he incorrectly simplified the second term.
4. Corrected Step 2
The correct Step 2 should be:
$8 x^6 \cdot 2 \cdot 5 \cdot x^6 \sqrt{2 x} \div 2 \cdot 4 \cdot x^5 \cdot x^3 \sqrt{2 x} = 80 x^{12} \sqrt{2x} \div 8 x^8 \sqrt{2x}$
5. Conclusion
Therefore, Seth's first mistake was made in Step 2, where he incorrectly simplified the second term.
### Examples
Simplifying radical expressions is useful in many areas of mathematics and physics. For example, when calculating the distance between two points in a coordinate plane, you often end up with a radical expression that needs to be simplified. Similarly, in physics, when dealing with energy or momentum calculations, simplifying radicals can make the calculations easier and more manageable. This skill is also crucial in engineering for structural analysis and design, where precise calculations are essential.
### Explanation
1. Problem Analysis
We are asked to identify the first mistake in Seth's simplification of the expression $8 x^6
\sqrt{200 x^{13}}
\div 2 x^5
\sqrt{32 x^7}$. We will analyze each step to find the error.
2. Step 1 Analysis
Step 1: $8 x^6
\sqrt{200 x^{13}}
\div 2 x^5
\sqrt{32 x^7} = 8 x^6
\sqrt{4 \cdot 25 \cdot 2 \cdot(x^6)^2 \cdot x}
\div 2 x^5
\sqrt{16 \cdot 2 \cdot(x^3)^2 \cdot x}$.
This step correctly factors the numbers and the powers of x inside the square roots.
3. Step 2 Analysis
Step 2: $8 x^6
\sqrt{4 \cdot 25 \cdot 2 \cdot(x^6)^2 \cdot x} = 8 x^6
\sqrt{100 \cdot 2 \cdot x^{12} \cdot x} = 8 x^6 \cdot 10 x^6
\sqrt{2x} = 80 x^{12}
\sqrt{2x}$. This part is correct.
Also, $2 x^5
\sqrt{16 \cdot 2 \cdot(x^3)^2 \cdot x} = 2 x^5
\sqrt{16 \cdot 2 \cdot x^6 \cdot x} = 2 x^5 \cdot 4 x^3
\sqrt{2x} = 8 x^8
\sqrt{2x}$.
Seth wrote $2 \cdot 16 \cdot x^5 \cdot x^3
\sqrt{2 x}$ which is incorrect. The 16 should be a 4, and the product should be $8x^8$ and not $32x^8$. Therefore, the first mistake was in Step 2, where he incorrectly simplified the second term.
4. Corrected Step 2
The correct Step 2 should be:
$8 x^6 \cdot 2 \cdot 5 \cdot x^6 \sqrt{2 x} \div 2 \cdot 4 \cdot x^5 \cdot x^3 \sqrt{2 x} = 80 x^{12} \sqrt{2x} \div 8 x^8 \sqrt{2x}$
5. Conclusion
Therefore, Seth's first mistake was made in Step 2, where he incorrectly simplified the second term.
### Examples
Simplifying radical expressions is useful in many areas of mathematics and physics. For example, when calculating the distance between two points in a coordinate plane, you often end up with a radical expression that needs to be simplified. Similarly, in physics, when dealing with energy or momentum calculations, simplifying radicals can make the calculations easier and more manageable. This skill is also crucial in engineering for structural analysis and design, where precise calculations are essential.
