Answer :
Let's carefully examine the steps provided for simplifying the given expression:
Given expression:
[tex]\[ 8x^6 \sqrt{200x^{13}} \div 2x^5 \sqrt{32x^7} \][/tex]
### Step 1:
[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]
In Step 1, Seth factored the numbers inside the square roots. Let's review this factorization:
- [tex]\( 200 = 4 \cdot 25 \cdot 2 \)[/tex]
- [tex]\( x^{13} = (x^6)^2 \cdot x \)[/tex]
- [tex]\( 32 = 16 \cdot 2 \)[/tex]
- [tex]\( x^7 = (x^3)^2 \cdot x \)[/tex]
This factorization is correct. The next step is simplifying these expressions.
### Step 2:
[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 Step 2, Seth separated the constants and variables, but notice:
- [tex]\( \sqrt{200x^{13}} = 10x^6\sqrt{2x} \)[/tex], not [tex]\( 2 \cdot 5 \cdot x^6 x^6\sqrt{2x} \)[/tex]
- Similarly, [tex]\( \sqrt{32x^7} = 4x^3\sqrt{2x} \)[/tex]
Therefore, Step 2 should correctly be:
[tex]\[ 8x^6 \cdot 10x^6 \sqrt{2x} \div 2x^5 \cdot 4x^3 \sqrt{2x} \][/tex]
### Correct Simplification:
[tex]\[ 80x^{12} \sqrt{2x} \div 8x^8 \sqrt{2x} \][/tex]
### Step 3:
[tex]\[ 80x^{12} \sqrt{2x} \div 32x^8 \sqrt{2x} \][/tex]
Simplifies to:
[tex]\[ \frac{80x^{12} \sqrt{2x}}{32x^8 \sqrt{2x}} \][/tex]
### Step 4:
[tex]\[ \frac{80x^{12}}{32x^8} \][/tex]
Dividing and simplifying:
[tex]\[ \frac{80}{32} \cdot \frac{x^{12}}{x^8} = 2.5x^4 = \frac{5}{2}x^4 \][/tex]
Thus, Seth's first mistake was made in Step 1, where he separated the constants and variables incorrectly.
So, the final answer is:
Seth's first mistake was made in Step 1, where he separated the constants and variables incorrectly.
Given expression:
[tex]\[ 8x^6 \sqrt{200x^{13}} \div 2x^5 \sqrt{32x^7} \][/tex]
### Step 1:
[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]
In Step 1, Seth factored the numbers inside the square roots. Let's review this factorization:
- [tex]\( 200 = 4 \cdot 25 \cdot 2 \)[/tex]
- [tex]\( x^{13} = (x^6)^2 \cdot x \)[/tex]
- [tex]\( 32 = 16 \cdot 2 \)[/tex]
- [tex]\( x^7 = (x^3)^2 \cdot x \)[/tex]
This factorization is correct. The next step is simplifying these expressions.
### Step 2:
[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 Step 2, Seth separated the constants and variables, but notice:
- [tex]\( \sqrt{200x^{13}} = 10x^6\sqrt{2x} \)[/tex], not [tex]\( 2 \cdot 5 \cdot x^6 x^6\sqrt{2x} \)[/tex]
- Similarly, [tex]\( \sqrt{32x^7} = 4x^3\sqrt{2x} \)[/tex]
Therefore, Step 2 should correctly be:
[tex]\[ 8x^6 \cdot 10x^6 \sqrt{2x} \div 2x^5 \cdot 4x^3 \sqrt{2x} \][/tex]
### Correct Simplification:
[tex]\[ 80x^{12} \sqrt{2x} \div 8x^8 \sqrt{2x} \][/tex]
### Step 3:
[tex]\[ 80x^{12} \sqrt{2x} \div 32x^8 \sqrt{2x} \][/tex]
Simplifies to:
[tex]\[ \frac{80x^{12} \sqrt{2x}}{32x^8 \sqrt{2x}} \][/tex]
### Step 4:
[tex]\[ \frac{80x^{12}}{32x^8} \][/tex]
Dividing and simplifying:
[tex]\[ \frac{80}{32} \cdot \frac{x^{12}}{x^8} = 2.5x^4 = \frac{5}{2}x^4 \][/tex]
Thus, Seth's first mistake was made in Step 1, where he separated the constants and variables incorrectly.
So, the final answer is:
Seth's first mistake was made in Step 1, where he separated the constants and variables incorrectly.
