Sure, let's solve the given expression step-by-step.
Given expression:
[tex]\[ \sqrt[4]{64 x^7 y^8} \div \sqrt[4]{4 x^3 y^{-4}} \][/tex]
First, we can break this expression down into more manageable parts:
1. Evaluate the Numerator: [tex]\[ \sqrt[4]{64 x^7 y^8} \][/tex]
- We know that [tex]\( \sqrt[4]{64} = (64)^{\frac{1}{4}} \)[/tex].
- Since [tex]\( 64 = 2^6 \)[/tex], we have [tex]\( (64)^{\frac{1}{4}} = (2^6)^{\frac{1}{4}} = 2^{6 \cdot \frac{1}{4}} = 2^{1.5} = 2 \cdot \sqrt{2} \)[/tex].
- For the [tex]\( x \)[/tex] term: [tex]\( \sqrt[4]{x^7} = x^{\frac{7}{4}} \)[/tex].
- For the [tex]\( y \)[/tex] term: [tex]\( \sqrt[4]{y^8} = (y^8)^{\frac{1}{4}} = y^{2} \)[/tex].
Combining these, we get:
[tex]\[ \sqrt[4]{64 x^7 y^8} = 2 \sqrt{2} \cdot x^{7/4} \cdot y^{2} \][/tex]
2. Evaluate the Denominator: [tex]\[ \sqrt[4]{4 x^3 y^{-4}} \][/tex]
- We know that [tex]\( \sqrt[4]{4} = (4)^{\frac{1}{4}} = (2^2)^{\frac{1}{4}} = 2^{0.5} = \sqrt{2} \)[/tex].
- For the [tex]\( x \)[/tex] term: [tex]\( \sqrt[4]{x^3} = x^{\frac{3}{4}} \)[/tex].
- For the [tex]\( y \)[/tex] term: [tex]\( \sqrt[4]{y^{-4}} = (y^{-4})^{\frac{1}{4}} = y^{-1} \)[/tex].
Combining these, we get:
[tex]\[ \sqrt[4]{4 x^3 y^{-4}} = \sqrt{2} \cdot x^{3/4} \cdot y^{-1} \][/tex]
3. Divide the Numerator by the Denominator:
- We have:
[tex]\[ \frac{\sqrt[4]{64 x^7 y^8}}{\sqrt[4]{4 x^3 y^{-4}}} = \frac{2\sqrt{2} \cdot x^{7/4} \cdot y^2}{\sqrt{2} \cdot x^{3/4} \cdot y^{-1}} \][/tex]
- Simplify the coefficients:
[tex]\[ \frac{2 \sqrt{2}}{\sqrt{2}} = 2 \][/tex]
- Simplify the [tex]\( x \)[/tex] terms:
[tex]\[ \frac{x^{7/4}}{x^{3/4}} = x^{(7/4) - (3/4)} = x^{4/4} = x \][/tex]
- Simplify the [tex]\( y \)[/tex] terms:
[tex]\[ \frac{y^2}{y^{-1}} = y^{2 - (-1)} = y^{3} \][/tex]
Combining these simplified parts, we get the final result:
[tex]\[ \frac{\sqrt[4]{64 x^7 y^8}}{\sqrt[4]{4 x^3 y^{-4}}} = 2 \cdot x \cdot y^3 \][/tex]
Thus, the detailed solution yields:
[tex]\[ 2 \cdot x \cdot y^3 \][/tex]
