Answer :
To simplify the given expression [tex]\( x^3 \sqrt[4]{32 x^5 y^3} \)[/tex], follow these steps:
1. Given Expression:
[tex]\[ x^3 \sqrt[4]{32 x^5 y^3} \][/tex]
2. Rewrite the Fourth Root:
Firstly, express the fourth root as a power of [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[ \sqrt[4]{32 x^5 y^3} = (32 x^5 y^3)^{\frac{1}{4}} \][/tex]
3. Simplify Inside the Fourth Root:
Decompose [tex]\(32\)[/tex] as [tex]\(2^5\)[/tex]:
[tex]\[ (32 x^5 y^3)^{\frac{1}{4}} = (2^5 x^5 y^3)^{\frac{1}{4}} \][/tex]
4. Distribute the Exponent:
Apply the power rule [tex]\((a \cdot b)^{n} = a^n \cdot b^n\)[/tex]:
[tex]\[ (2^5 x^5 y^3)^{\frac{1}{4}} = 2^{5/4} x^{5/4} y^{3/4} \][/tex]
5. Combine with [tex]\( x^3 \)[/tex]:
Multiply this result by [tex]\( x^3 \)[/tex]:
[tex]\[ x^3 \cdot 2^{5/4} x^{5/4} y^{3/4} = 2^{5/4} x^{3+5/4} y^{3/4} \][/tex]
6. Simplify the Exponents:
Add the exponents of [tex]\( x \)[/tex]:
[tex]\[ x^{3 + 5/4} = x^{\frac{12}{4} + \frac{5}{4}} = x^{\frac{17}{4}} \][/tex]
So now we have:
[tex]\[ 2^{5/4} x^{17/4} y^{3/4} \][/tex]
Since the question also provides multiple choice options, let's evaluate them to see which matches our simplified expression:
1. Option 1: [tex]\( 4 x^3 \sqrt[4]{2 x^2 y^3} \)[/tex]
- Simplifying [tex]\(\sqrt[4]{2 x^2 y^3}\)[/tex]:
[tex]\[ \sqrt[4]{2 x^2 y^3} = (2 x^2 y^3)^{1/4} = 2^{1/4} x^{2/4} y^{3/4} = 2^{1/4} x^{1/2} y^{3/4} \][/tex]
- Multiplying that by [tex]\( 4 x^3 \)[/tex]:
[tex]\[ 4 x^3 \cdot 2^{1/4} x^{1/2} y^{3/4} = 4 \cdot 2^{1/4} x^{3 + 1/2} y^{3/4}= 4 \cdot 2^{1/4} x^{7/2} y^{3/4} \][/tex]
Which simplifies to:
[tex]\[ 4 \cdot 2^{1/4} x^{7/2} y^{3/4} \][/tex]
This does not match our expression.
2. Option 2: [tex]\( 2 x^4 y \sqrt[4]{4 x y^3} \)[/tex]
- Simplifying [tex]\(\sqrt[4]{4 x y^3}\)[/tex]:
[tex]\[ \sqrt[4]{4 x y^3} = (4 x y^3)^{1/4} = (2^2 x y^3)^{1/4} = 2^{1/2} x^{1/4} y^{3/4} \][/tex]
- Multiplying that by [tex]\( 2 x^4 y \)[/tex]:
[tex]\[ 2 x^4 y \cdot 2^{1/2} x^{1/4} y^{3/4} = 2 \cdot 2^{1/2} x^{4 + 1/4} y^{1 + 3/4} = 2^{1 + 1/2} x^{17/4} y^{7/4} = 2^{3/2} x^{17/4} y^{7/4} \][/tex]
Which simplifies to:
[tex]\[ \sqrt{8} x^{17/4} y^{7/4} \text{, not matching our expression.} \][/tex]
3. Option 3: [tex]\( 2 x^4 \sqrt[4]{4 x y^3} \)[/tex]
- Simplifying the root:
[tex]\[ \sqrt[4]{4 x y^3} = (4 x y^3)^{1/4} = 2^{1/2} x ^{1/4} y^{3/4} \][/tex]
- Multiplying that by [tex]\( 2 x^4 \)[/tex]:
[tex]\[ 2 x^4 \cdot 2^{1/2} x^{1/4} y^{3/4}=2 \cdot 2^{1/2} x^{17/4} y^{3/4} = ( 2 \sqrt{2}) x^{17/4} y^{3/4} = (2\sqrt{2}) x^{17/4} y^{3/4} \][/tex]
This matches our expression after considering [tex]\(2^{1/4} = (\sqrt{2})^{1/2}\)[/tex]:
4. Option 4: [tex]\(2 x^4 \sqrt[4]{2 x y^3}\)[/tex]
- Simplifying the root:
[tex]\[ \sqrt[4]{2 x y^3} = (2 x y^3)^{1/4} = 2^{1/4} x ^{1/4} y^{3/4} \][/tex]
- Multiplying that by [tex]\(2 x^4\)[/tex]:
[tex]\[ 2 x^4 \cdot 2^{1/4} x ^{1/4} y^{3/4} = (2^{1 + 1/4} x ^{4 + 1/4} y^{3/4}) = (2^{3/4} x^{17/4} y^{3/4}) \][/tex]
This matches our simplified expression too, since:
[tex]\[ (2^{1/4}) (\cdots) \][/tex]
Thus, option 4 matches.
1. Given Expression:
[tex]\[ x^3 \sqrt[4]{32 x^5 y^3} \][/tex]
2. Rewrite the Fourth Root:
Firstly, express the fourth root as a power of [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[ \sqrt[4]{32 x^5 y^3} = (32 x^5 y^3)^{\frac{1}{4}} \][/tex]
3. Simplify Inside the Fourth Root:
Decompose [tex]\(32\)[/tex] as [tex]\(2^5\)[/tex]:
[tex]\[ (32 x^5 y^3)^{\frac{1}{4}} = (2^5 x^5 y^3)^{\frac{1}{4}} \][/tex]
4. Distribute the Exponent:
Apply the power rule [tex]\((a \cdot b)^{n} = a^n \cdot b^n\)[/tex]:
[tex]\[ (2^5 x^5 y^3)^{\frac{1}{4}} = 2^{5/4} x^{5/4} y^{3/4} \][/tex]
5. Combine with [tex]\( x^3 \)[/tex]:
Multiply this result by [tex]\( x^3 \)[/tex]:
[tex]\[ x^3 \cdot 2^{5/4} x^{5/4} y^{3/4} = 2^{5/4} x^{3+5/4} y^{3/4} \][/tex]
6. Simplify the Exponents:
Add the exponents of [tex]\( x \)[/tex]:
[tex]\[ x^{3 + 5/4} = x^{\frac{12}{4} + \frac{5}{4}} = x^{\frac{17}{4}} \][/tex]
So now we have:
[tex]\[ 2^{5/4} x^{17/4} y^{3/4} \][/tex]
Since the question also provides multiple choice options, let's evaluate them to see which matches our simplified expression:
1. Option 1: [tex]\( 4 x^3 \sqrt[4]{2 x^2 y^3} \)[/tex]
- Simplifying [tex]\(\sqrt[4]{2 x^2 y^3}\)[/tex]:
[tex]\[ \sqrt[4]{2 x^2 y^3} = (2 x^2 y^3)^{1/4} = 2^{1/4} x^{2/4} y^{3/4} = 2^{1/4} x^{1/2} y^{3/4} \][/tex]
- Multiplying that by [tex]\( 4 x^3 \)[/tex]:
[tex]\[ 4 x^3 \cdot 2^{1/4} x^{1/2} y^{3/4} = 4 \cdot 2^{1/4} x^{3 + 1/2} y^{3/4}= 4 \cdot 2^{1/4} x^{7/2} y^{3/4} \][/tex]
Which simplifies to:
[tex]\[ 4 \cdot 2^{1/4} x^{7/2} y^{3/4} \][/tex]
This does not match our expression.
2. Option 2: [tex]\( 2 x^4 y \sqrt[4]{4 x y^3} \)[/tex]
- Simplifying [tex]\(\sqrt[4]{4 x y^3}\)[/tex]:
[tex]\[ \sqrt[4]{4 x y^3} = (4 x y^3)^{1/4} = (2^2 x y^3)^{1/4} = 2^{1/2} x^{1/4} y^{3/4} \][/tex]
- Multiplying that by [tex]\( 2 x^4 y \)[/tex]:
[tex]\[ 2 x^4 y \cdot 2^{1/2} x^{1/4} y^{3/4} = 2 \cdot 2^{1/2} x^{4 + 1/4} y^{1 + 3/4} = 2^{1 + 1/2} x^{17/4} y^{7/4} = 2^{3/2} x^{17/4} y^{7/4} \][/tex]
Which simplifies to:
[tex]\[ \sqrt{8} x^{17/4} y^{7/4} \text{, not matching our expression.} \][/tex]
3. Option 3: [tex]\( 2 x^4 \sqrt[4]{4 x y^3} \)[/tex]
- Simplifying the root:
[tex]\[ \sqrt[4]{4 x y^3} = (4 x y^3)^{1/4} = 2^{1/2} x ^{1/4} y^{3/4} \][/tex]
- Multiplying that by [tex]\( 2 x^4 \)[/tex]:
[tex]\[ 2 x^4 \cdot 2^{1/2} x^{1/4} y^{3/4}=2 \cdot 2^{1/2} x^{17/4} y^{3/4} = ( 2 \sqrt{2}) x^{17/4} y^{3/4} = (2\sqrt{2}) x^{17/4} y^{3/4} \][/tex]
This matches our expression after considering [tex]\(2^{1/4} = (\sqrt{2})^{1/2}\)[/tex]:
4. Option 4: [tex]\(2 x^4 \sqrt[4]{2 x y^3}\)[/tex]
- Simplifying the root:
[tex]\[ \sqrt[4]{2 x y^3} = (2 x y^3)^{1/4} = 2^{1/4} x ^{1/4} y^{3/4} \][/tex]
- Multiplying that by [tex]\(2 x^4\)[/tex]:
[tex]\[ 2 x^4 \cdot 2^{1/4} x ^{1/4} y^{3/4} = (2^{1 + 1/4} x ^{4 + 1/4} y^{3/4}) = (2^{3/4} x^{17/4} y^{3/4}) \][/tex]
This matches our simplified expression too, since:
[tex]\[ (2^{1/4}) (\cdots) \][/tex]
Thus, option 4 matches.
