Answer :
Sure! Let's go through the problem step-by-step to identify where Seth made a mistake in rewriting and simplifying the expression.
We are given the expression:
[tex]\[ 8 x^6 \sqrt{200 x^{13}} \div 2 x^5 \sqrt{32 x^7} \][/tex]
Seth’s steps are as follows:
1. Step 1 attempts to factor the terms inside the square roots:
[tex]\[ 8 x^6 \sqrt{4 \cdot 25 \cdot 2 \cdot (x^6)^2 \cdot x} \div 2 x^5 \sqrt{16 \cdot 2 \cdot (x^3)^2 \cdot x} \][/tex]
2. Step 2 simplifies the expression:
[tex]\[ 8 \cdot 2 \cdot 5 \cdot x^6 \cdot x^6 \sqrt{2 x} \div 2 \cdot 16 \cdot x^5 \cdot x^3 \sqrt{2 x} \][/tex]
Here, Seth has made his first mistake. In this step, the multiplication and simplification within the expression should have been aligned correctly. Let's break this step down:
- The expression inside the first square root should have resulted in extracting the factors 4 (as 2), 25 (as 5), and [tex]\( (x^6)^2 \)[/tex] (as [tex]\( x^6 \)[/tex]), but Seth simplified incorrectly. Correct factors: [tex]\( 4 \cdot 25 = 100 \)[/tex] which should come out as 10, not 2 and 5 separately.
- Similarly, in the denominator, he simplified the square root expression incorrectly.
3. Step 3 rewrites the simplified expression:
[tex]\[ 80 x^{12} \sqrt{2 x} \div 32 x^8 \sqrt{2 x} \][/tex]
4. Step 4 expresses the division as a fraction:
[tex]\[ \frac{80 x^{12} \sqrt{2 x}}{32 x^8 \sqrt{2 x}} \][/tex]
5. Step 5 simplifies the fraction:
[tex]\[ \frac{5}{2} x^4 \][/tex]
Therefore, the first mistake occurred in Step 2, where Seth incorrectly simplified and multiplied the factors. After fixing these, the simplification process would have led to consistent results with the mathematical operations of the square roots and products properly handled.
We are given the expression:
[tex]\[ 8 x^6 \sqrt{200 x^{13}} \div 2 x^5 \sqrt{32 x^7} \][/tex]
Seth’s steps are as follows:
1. Step 1 attempts to factor the terms inside the square roots:
[tex]\[ 8 x^6 \sqrt{4 \cdot 25 \cdot 2 \cdot (x^6)^2 \cdot x} \div 2 x^5 \sqrt{16 \cdot 2 \cdot (x^3)^2 \cdot x} \][/tex]
2. Step 2 simplifies the expression:
[tex]\[ 8 \cdot 2 \cdot 5 \cdot x^6 \cdot x^6 \sqrt{2 x} \div 2 \cdot 16 \cdot x^5 \cdot x^3 \sqrt{2 x} \][/tex]
Here, Seth has made his first mistake. In this step, the multiplication and simplification within the expression should have been aligned correctly. Let's break this step down:
- The expression inside the first square root should have resulted in extracting the factors 4 (as 2), 25 (as 5), and [tex]\( (x^6)^2 \)[/tex] (as [tex]\( x^6 \)[/tex]), but Seth simplified incorrectly. Correct factors: [tex]\( 4 \cdot 25 = 100 \)[/tex] which should come out as 10, not 2 and 5 separately.
- Similarly, in the denominator, he simplified the square root expression incorrectly.
3. Step 3 rewrites the simplified expression:
[tex]\[ 80 x^{12} \sqrt{2 x} \div 32 x^8 \sqrt{2 x} \][/tex]
4. Step 4 expresses the division as a fraction:
[tex]\[ \frac{80 x^{12} \sqrt{2 x}}{32 x^8 \sqrt{2 x}} \][/tex]
5. Step 5 simplifies the fraction:
[tex]\[ \frac{5}{2} x^4 \][/tex]
Therefore, the first mistake occurred in Step 2, where Seth incorrectly simplified and multiplied the factors. After fixing these, the simplification process would have led to consistent results with the mathematical operations of the square roots and products properly handled.
