Answer :
To find Seth's mistake, let's review each step of his process:
Given the expression:
[tex]\[ 8x^6 \sqrt{200x^{13}} \div 2x^5 \sqrt{32x^7} \][/tex]
Step 1: Simplify inside the square roots.
Breaking down the square roots:
[tex]\[ \sqrt{200x^{13}} = \sqrt{4 \cdot 25 \cdot 2 \cdot (x^6)^2 \cdot x} \][/tex]
[tex]\[ \sqrt{32x^7} = \sqrt{16 \cdot 2 \cdot (x^3)^2 \cdot x} \][/tex]
So, step 1 is correct.
Step 2: Simplify the square roots into products.
[tex]\[ \sqrt{200x^{13}} = 2 \cdot 5 \cdot x^6 \sqrt{2x} \][/tex]
[tex]\[ \sqrt{32x^7} = 4 \cdot x^3 \sqrt{2x} \][/tex]
The expression becomes:
[tex]\[ 8 \times 2 \times 5 \times x^6 \times x^6 \times \sqrt{2x} \div 2 \times 4 \times x^5 \times x^3 \times \sqrt{2x} \][/tex]
So, step 2 is correct.
Step 3: Combine and simplify terms.
The expression from step 2 is:
[tex]\[ 80 x^{12} \sqrt{2x} \div 32 x^8 \sqrt{2x} \][/tex]
Here, Seth made his first mistake. He should have divided each part by the correct term.
- Coefficients: [tex]\( \frac{80}{32} = \frac{5}{2} \)[/tex]
- Powers of [tex]\( x \)[/tex]: [tex]\( x^{12} \div x^8 = x^{4} \)[/tex]
- [tex]\( \sqrt{2x} \div \sqrt{2x} = 1 \)[/tex]
Thus, the entire expression should simplify to:
[tex]\[ \frac{5}{2} x^{4} \][/tex]
However, Seth moved to step 4 using an incorrect intermediary step. Thus, the mistake is made in step 3 where he did not divide correctly.
Step 4 and Step 5: Continuing despite the mistake
Continuing to step 4 and 5, Seth calculates:
In step 4, he mistakenly writes [tex]\( x^6 \)[/tex] after dividing, which was incorrect.
This leads to the final result:
[tex]\[ \frac{5}{2} x^4 \][/tex]
The first encountered mistake is in step 3 where the division was not correctly expressed. The correct simplification for step 3 should have continued with:
[tex]\[ \frac{5}{2} x^4 \][/tex]
Given the expression:
[tex]\[ 8x^6 \sqrt{200x^{13}} \div 2x^5 \sqrt{32x^7} \][/tex]
Step 1: Simplify inside the square roots.
Breaking down the square roots:
[tex]\[ \sqrt{200x^{13}} = \sqrt{4 \cdot 25 \cdot 2 \cdot (x^6)^2 \cdot x} \][/tex]
[tex]\[ \sqrt{32x^7} = \sqrt{16 \cdot 2 \cdot (x^3)^2 \cdot x} \][/tex]
So, step 1 is correct.
Step 2: Simplify the square roots into products.
[tex]\[ \sqrt{200x^{13}} = 2 \cdot 5 \cdot x^6 \sqrt{2x} \][/tex]
[tex]\[ \sqrt{32x^7} = 4 \cdot x^3 \sqrt{2x} \][/tex]
The expression becomes:
[tex]\[ 8 \times 2 \times 5 \times x^6 \times x^6 \times \sqrt{2x} \div 2 \times 4 \times x^5 \times x^3 \times \sqrt{2x} \][/tex]
So, step 2 is correct.
Step 3: Combine and simplify terms.
The expression from step 2 is:
[tex]\[ 80 x^{12} \sqrt{2x} \div 32 x^8 \sqrt{2x} \][/tex]
Here, Seth made his first mistake. He should have divided each part by the correct term.
- Coefficients: [tex]\( \frac{80}{32} = \frac{5}{2} \)[/tex]
- Powers of [tex]\( x \)[/tex]: [tex]\( x^{12} \div x^8 = x^{4} \)[/tex]
- [tex]\( \sqrt{2x} \div \sqrt{2x} = 1 \)[/tex]
Thus, the entire expression should simplify to:
[tex]\[ \frac{5}{2} x^{4} \][/tex]
However, Seth moved to step 4 using an incorrect intermediary step. Thus, the mistake is made in step 3 where he did not divide correctly.
Step 4 and Step 5: Continuing despite the mistake
Continuing to step 4 and 5, Seth calculates:
In step 4, he mistakenly writes [tex]\( x^6 \)[/tex] after dividing, which was incorrect.
This leads to the final result:
[tex]\[ \frac{5}{2} x^4 \][/tex]
The first encountered mistake is in step 3 where the division was not correctly expressed. The correct simplification for step 3 should have continued with:
[tex]\[ \frac{5}{2} x^4 \][/tex]
