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Answer :
- Step 1 correctly factors the terms inside the square roots.
- Step 2 incorrectly simplifies $\sqrt{16}$ as $16$ instead of $4$.
- The mistake in Step 2 leads to an incorrect expression.
- Seth's first mistake was made in Step 2, where he incorrectly simplified the square root. The answer is $\boxed{Step 2, incorrectly simplified the square root}$
### Explanation
1. Analyzing the Problem
We are given the expression $8 x^6 \sqrt{200 x^{13}} \div 2 x^5 \sqrt{32 x^7}$ and Seth's attempt to simplify it. Our goal is to identify the first step where Seth made a mistake.
2. Checking Step 1
Step 1: $8 x^6 \sqrt{200 x^{13}} \div 2 x^5 \sqrt{32 x^7} = 8 x^6 \sqrt{4 \cdot 25 \cdot 2 \cdot(x^6)^2 \cdot x} \div 2 x^5 \sqrt{16 \cdot 2 \cdot(x^3)^2 \cdot x}$.
Here, $200x^{13}$ is correctly factored as $4 \cdot 25 \cdot 2 \cdot (x^6)^2 \cdot x = 200x^{12} \cdot x = 200x^{13}$. Also, $32x^7$ is correctly factored as $16 \cdot 2 \cdot (x^3)^2 \cdot x = 32x^6 \cdot x = 32x^7$. So, Step 1 is correct.
3. Checking Step 2
Step 2: $8 x^6 \sqrt{4 \cdot 25 \cdot 2 \cdot(x^6)^2 \cdot x} \div 2 x^5 \sqrt{16 \cdot 2 \cdot(x^3)^2 \cdot x} = 8 \cdot 2 \cdot 5 \cdot x^6 \cdot x^6 \sqrt{2 x} \div 2 \cdot 4 \cdot x^5 \cdot x^3 \sqrt{2 x}$.
In the first term, $\sqrt{4 \cdot 25 \cdot 2 \cdot(x^6)^2 \cdot x} = 2 \cdot 5 \cdot x^6 \sqrt{2x} = 10x^6\sqrt{2x}$. So, $8x^6 \sqrt{200x^{13}} = 8x^6 \cdot 10x^6\sqrt{2x} = 80x^{12}\sqrt{2x}$.
In the second term, $\sqrt{16 \cdot 2 \cdot(x^3)^2 \cdot x} = 4 \cdot x^3 \sqrt{2x}$. So, $2x^5 \sqrt{32x^7} = 2x^5 \cdot 4x^3\sqrt{2x} = 8x^8\sqrt{2x}$.
Seth wrote $2 \cdot 16 \cdot x^5 \cdot x^3 \sqrt{2 x}$ instead of $2 \cdot 4 \cdot x^5 \cdot x^3 \sqrt{2 x}$. Therefore, Seth incorrectly simplified $\sqrt{16}$ as $16$ instead of $4$.
4. Conclusion
Therefore, Seth's first mistake was made in Step 2, where he incorrectly simplified $\sqrt{16}$ as $16$ instead of $4$.
### Examples
When simplifying expressions with radicals, it's crucial to correctly identify and extract perfect squares. For instance, in construction, calculating the area of a square room with sides involving square roots requires accurate simplification to determine the amount of flooring needed. A mistake in simplifying radicals can lead to incorrect area calculations, resulting in either insufficient or excessive material purchases.
- Step 2 incorrectly simplifies $\sqrt{16}$ as $16$ instead of $4$.
- The mistake in Step 2 leads to an incorrect expression.
- Seth's first mistake was made in Step 2, where he incorrectly simplified the square root. The answer is $\boxed{Step 2, incorrectly simplified the square root}$
### Explanation
1. Analyzing the Problem
We are given the expression $8 x^6 \sqrt{200 x^{13}} \div 2 x^5 \sqrt{32 x^7}$ and Seth's attempt to simplify it. Our goal is to identify the first step where Seth made a mistake.
2. Checking Step 1
Step 1: $8 x^6 \sqrt{200 x^{13}} \div 2 x^5 \sqrt{32 x^7} = 8 x^6 \sqrt{4 \cdot 25 \cdot 2 \cdot(x^6)^2 \cdot x} \div 2 x^5 \sqrt{16 \cdot 2 \cdot(x^3)^2 \cdot x}$.
Here, $200x^{13}$ is correctly factored as $4 \cdot 25 \cdot 2 \cdot (x^6)^2 \cdot x = 200x^{12} \cdot x = 200x^{13}$. Also, $32x^7$ is correctly factored as $16 \cdot 2 \cdot (x^3)^2 \cdot x = 32x^6 \cdot x = 32x^7$. So, Step 1 is correct.
3. Checking Step 2
Step 2: $8 x^6 \sqrt{4 \cdot 25 \cdot 2 \cdot(x^6)^2 \cdot x} \div 2 x^5 \sqrt{16 \cdot 2 \cdot(x^3)^2 \cdot x} = 8 \cdot 2 \cdot 5 \cdot x^6 \cdot x^6 \sqrt{2 x} \div 2 \cdot 4 \cdot x^5 \cdot x^3 \sqrt{2 x}$.
In the first term, $\sqrt{4 \cdot 25 \cdot 2 \cdot(x^6)^2 \cdot x} = 2 \cdot 5 \cdot x^6 \sqrt{2x} = 10x^6\sqrt{2x}$. So, $8x^6 \sqrt{200x^{13}} = 8x^6 \cdot 10x^6\sqrt{2x} = 80x^{12}\sqrt{2x}$.
In the second term, $\sqrt{16 \cdot 2 \cdot(x^3)^2 \cdot x} = 4 \cdot x^3 \sqrt{2x}$. So, $2x^5 \sqrt{32x^7} = 2x^5 \cdot 4x^3\sqrt{2x} = 8x^8\sqrt{2x}$.
Seth wrote $2 \cdot 16 \cdot x^5 \cdot x^3 \sqrt{2 x}$ instead of $2 \cdot 4 \cdot x^5 \cdot x^3 \sqrt{2 x}$. Therefore, Seth incorrectly simplified $\sqrt{16}$ as $16$ instead of $4$.
4. Conclusion
Therefore, Seth's first mistake was made in Step 2, where he incorrectly simplified $\sqrt{16}$ as $16$ instead of $4$.
### Examples
When simplifying expressions with radicals, it's crucial to correctly identify and extract perfect squares. For instance, in construction, calculating the area of a square room with sides involving square roots requires accurate simplification to determine the amount of flooring needed. A mistake in simplifying radicals can lead to incorrect area calculations, resulting in either insufficient or excessive material purchases.
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