Question: $1$ $\newline$ Simplify the following according to the roles of exponents. $\newline$ $\frac{\sqrt{x^{3} y^{5}} \sqrt{x y^{3}}}{\sqrt[4]{x^{7} y^{2}} 4 \sqrt{x y^{6}}} .$

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Q. Question:

1

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Simplify the following according to the roles of exponents.

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\frac{\sqrt{x^{3} y^{5}} \sqrt{x y^{3}}}{\sqrt[4]{x^{7} y^{2}} 4 \sqrt{x y^{6}}} .

Combine square roots: Combine the square roots: $\sqrt{x^{3}y^{5}} \times \sqrt{xy^{3}} = \sqrt{x^{3+1}y^{5+3}} = \sqrt{x^4y^8}.$
Combine roots in denominator: Combine the fourth root and the square root in the denominator: $\sqrt[4]{x^{7}y^{2}} \times 4\sqrt{xy^{6}} = \sqrt[4]{x^{7}y^{2}} \times \sqrt[4]{x^{2}y^{6\times2}} = \sqrt[4]{x^{7+2}y^{2+12}} = \sqrt[4]{x^{9}y^{14}}$ .
Convert to powers: Now we have: $\frac{\sqrt{x^4y^8}}{\sqrt[4]{x^9y^{14}}}$ .
Divide exponents: Convert the square root to a power of $\frac{1}{2}$ : $(x^4y^8)^{\frac{1}{2}} = x^{4*\frac{1}{2}}y^{8*\frac{1}{2}} = x^2y^4$ .
Convert negative exponents: Convert the fourth root to a power of $\frac{1}{4}$ : $(x^9y^{14})^{\frac{1}{4}} = x^{9\cdot(\frac{1}{4})}y^{14\cdot(\frac{1}{4})} = x^{\frac{9}{4}}y^{\frac{14}{4}} = x^{\frac{9}{4}}y^{\frac{7}{2}}$ .
Combine fractions: Divide the exponents: $(x^2y^4)/(x^{9/4}y^{7/2}) = x^{2-(9/4)}y^{4-(7/2)} = x^{8/4-9/4}y^{8/4-14/4} = x^{-1/4}y^{-6/4} = x^{-1/4}y^{-3/2}$ .
Combine fractions: Divide the exponents: $(x^2y^4)/(x^{9/4}y^{7/2}) = x^{2-(9/4)}y^{4-(7/2)} = x^{8/4-9/4}y^{8/4-14/4} = x^{-1/4}y^{-6/4} = x^{-1/4}y^{-3/2}$ . Since we can't have negative exponents in the final answer, we write them as fractions: $x^{-1/4} = 1/(x^{1/4})$ and $y^{-3/2} = 1/(y^{3/2})$ .
Combine fractions: Divide the exponents: $(x^2y^4)/(x^{9/4}y^{7/2}) = x^{2-(9/4)}y^{4-(7/2)} = x^{8/4-9/4}y^{8/4-14/4} = x^{-1/4}y^{-6/4} = x^{-1/4}y^{-3/2}$ .Since we can't have negative exponents in the final answer, we write them as fractions: $x^{-1/4} = 1/(x^{1/4})$ and $y^{-3/2} = 1/(y^{3/2})$ .Combine the fractions: $1/(x^{1/4}y^{3/2})$ .