Answer :
Certainly! Let's review Seth's steps and identify where he made a mistake in simplifying the expression:
The original expression provided is:
[tex]\[ 8 x^6 \sqrt{200 x^{13}} \div 2 x^5 \sqrt{32 x^7} \][/tex]
### Step-by-step Solution:
1. Factor Inside the Square Roots:
- Numerator's Square Root, [tex]\(\sqrt{200 x^{13}}\)[/tex]:
- [tex]\(200\)[/tex] can be factored into [tex]\(4 \times 25 \times 2\)[/tex].
- [tex]\(x^{13}\)[/tex] can be rewritten as [tex]\((x^6)^2 \times x\)[/tex].
- Denominator's Square Root, [tex]\(\sqrt{32 x^7}\)[/tex]:
- [tex]\(32\)[/tex] can be factored into [tex]\(16 \times 2\)[/tex].
- [tex]\(x^7\)[/tex] can be rewritten as [tex]\((x^3)^2 \times x\)[/tex].
So the expression becomes:
[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. Simplification of the Square Roots:
- Square root properties can be used: [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex].
- Take the square roots where applicable:
- Numerator: [tex]\(\sqrt{4} = 2\)[/tex], [tex]\(\sqrt{25} = 5\)[/tex], [tex]\(\sqrt{(x^6)^2} = x^6\)[/tex].
- Denominator: [tex]\(\sqrt{16} = 4\)[/tex], [tex]\(\sqrt{(x^3)^2} = x^3\)[/tex].
So the expression should simplify correctly to:
[tex]\[ 8 \cdot 2 \cdot 5 \cdot x^6 \cdot x^6 \sqrt{2x} \div 2 \cdot 4 \cdot x^5 \cdot x^3 \sqrt{2x} \][/tex]
3. Identify and Correct Coefficients Multiplication:
Seth made an error in this step:
- From [tex]\(8 \cdot 2 \cdot 5 = 80\)[/tex].
- From [tex]\(2 \cdot 4 \cdot 2 = 16\)[/tex], but Seth initially wrote incorrectly.
Correct multiplication should have given:
[tex]\[ 80 x^{12} \sqrt{2x} \div 32 x^8 \sqrt{2x} \][/tex]
4. Cancel Common Factors:
- Divide both the coefficients and any like terms using the properties of division:
- [tex]\(\frac{80}{32} = \frac{5}{2}\)[/tex].
- [tex]\(x^{12} \div x^8 = x^4\)[/tex].
- [tex]\(\sqrt{2x}\)[/tex] terms in numerator and denominator cancel out.
This simplifies to:
[tex]\[ \frac{5}{2} x^4 \][/tex]
Thus, Seth's mistake occurred in Step 2 while multiplying the coefficients incorrectly. After correction, the simplified expression is [tex]\(\frac{5}{2} x^4\)[/tex].
The original expression provided is:
[tex]\[ 8 x^6 \sqrt{200 x^{13}} \div 2 x^5 \sqrt{32 x^7} \][/tex]
### Step-by-step Solution:
1. Factor Inside the Square Roots:
- Numerator's Square Root, [tex]\(\sqrt{200 x^{13}}\)[/tex]:
- [tex]\(200\)[/tex] can be factored into [tex]\(4 \times 25 \times 2\)[/tex].
- [tex]\(x^{13}\)[/tex] can be rewritten as [tex]\((x^6)^2 \times x\)[/tex].
- Denominator's Square Root, [tex]\(\sqrt{32 x^7}\)[/tex]:
- [tex]\(32\)[/tex] can be factored into [tex]\(16 \times 2\)[/tex].
- [tex]\(x^7\)[/tex] can be rewritten as [tex]\((x^3)^2 \times x\)[/tex].
So the expression becomes:
[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. Simplification of the Square Roots:
- Square root properties can be used: [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex].
- Take the square roots where applicable:
- Numerator: [tex]\(\sqrt{4} = 2\)[/tex], [tex]\(\sqrt{25} = 5\)[/tex], [tex]\(\sqrt{(x^6)^2} = x^6\)[/tex].
- Denominator: [tex]\(\sqrt{16} = 4\)[/tex], [tex]\(\sqrt{(x^3)^2} = x^3\)[/tex].
So the expression should simplify correctly to:
[tex]\[ 8 \cdot 2 \cdot 5 \cdot x^6 \cdot x^6 \sqrt{2x} \div 2 \cdot 4 \cdot x^5 \cdot x^3 \sqrt{2x} \][/tex]
3. Identify and Correct Coefficients Multiplication:
Seth made an error in this step:
- From [tex]\(8 \cdot 2 \cdot 5 = 80\)[/tex].
- From [tex]\(2 \cdot 4 \cdot 2 = 16\)[/tex], but Seth initially wrote incorrectly.
Correct multiplication should have given:
[tex]\[ 80 x^{12} \sqrt{2x} \div 32 x^8 \sqrt{2x} \][/tex]
4. Cancel Common Factors:
- Divide both the coefficients and any like terms using the properties of division:
- [tex]\(\frac{80}{32} = \frac{5}{2}\)[/tex].
- [tex]\(x^{12} \div x^8 = x^4\)[/tex].
- [tex]\(\sqrt{2x}\)[/tex] terms in numerator and denominator cancel out.
This simplifies to:
[tex]\[ \frac{5}{2} x^4 \][/tex]
Thus, Seth's mistake occurred in Step 2 while multiplying the coefficients incorrectly. After correction, the simplified expression is [tex]\(\frac{5}{2} x^4\)[/tex].
