Sure, let's simplify the given algebraic expression step-by-step:
Given expression:
[tex]\[ \frac{x^4 y^2}{\sqrt[4]{x^3 y^2}} \][/tex]
First, recognize that the fourth root can be expressed as an exponent of [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[ \sqrt[4]{x^3 y^2} = (x^3 y^2)^{\frac{1}{4}} \][/tex]
So, our expression now looks like:
[tex]\[ \frac{x^4 y^2}{(x^3 y^2)^{\frac{1}{4}}} \][/tex]
Next, use the property of exponents that states [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[ (x^3 y^2)^{\frac{1}{4}} = x^{3 \cdot \frac{1}{4}} y^{2 \cdot \frac{1}{4}} = x^{\frac{3}{4}} y^{\frac{1}{2}} \][/tex]
Now, substitute this back into the fraction:
[tex]\[ \frac{x^4 y^2}{x^{\frac{3}{4}} y^{\frac{1}{2}}} \][/tex]
Next, simplify the fraction by applying the property of exponents [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex]:
[tex]\[ x^{4 - \frac{3}{4}} \cdot y^{2 - \frac{1}{2}} \][/tex]
Calculate each exponent:
[tex]\[ 4 - \frac{3}{4} = \frac{16}{4} - \frac{3}{4} = \frac{13}{4} \][/tex]
[tex]\[ 2 - \frac{1}{2} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2} \][/tex]
So, the expression simplifies to:
[tex]\[ x^{\frac{13}{4}} y^{\frac{3}{2}} \][/tex]
Finally, this can be further simplified or rearranged for readability. Recognizing [tex]\(x^{\frac{13}{4}} = x \cdot x^{\frac{9}{4}}\)[/tex]:
[tex]\[ x \cdot (x^{\frac{9}{4}} y^{\frac{3}{2}}) = x \cdot (x^{2.25} y^{1.5}) \][/tex]
Hence, the simplified form of the given expression is:
[tex]\[ x \cdot (x^3 y^2)^{\frac{3}{4}} \][/tex]
So, our final simplified expression is:
[tex]\[ x \cdot (x^3 y^2)^{\frac{3}{4}} \][/tex]
