Solve (frac{4X^{-1}/2X^{3}Y^{-3/4}}{6X^3/2Y^-4/2Y^4/3})^2

$$( \frac { 4 X ^ { - 1 } / 2 X ^ { 3 } Y ^ { - 3 / 4 } } { 6 X ^ { 3 / 2 } Y ^ { - 4 } / 2 Y ^ { 4 } / 3 } ) ^ { 2 }$$

Evaluate

$$\frac{4X}{Y^{\frac{3}{2}}}$$

Solution Steps

Divide 4X^{-1} by 2 to get 2X^{-1}.

$$\left(\frac{2X^{-1}X^{3}Y^{-\frac{3}{4}}}{\frac{\frac{6X^{\frac{3}{2}}Y^{-4}}{2}Y^{4}}{3}}\right)^{2}$$

To multiply powers of the same base, add their exponents. Add -1 and 3 to get 2.

$$\left(\frac{2X^{2}Y^{-\frac{3}{4}}}{\frac{\frac{6X^{\frac{3}{2}}Y^{-4}}{2}Y^{4}}{3}}\right)^{2}$$

Divide 6X^{\frac{3}{2}}Y^{-4} by 2 to get 3X^{\frac{3}{2}}Y^{-4}.

$$\left(\frac{2X^{2}Y^{-\frac{3}{4}}}{\frac{3X^{\frac{3}{2}}Y^{-4}Y^{4}}{3}}\right)^{2}$$

Multiply Y^{-4} and Y^{4} to get 1.

$$\left(\frac{2X^{2}Y^{-\frac{3}{4}}}{\frac{3X^{\frac{3}{2}}}{3}}\right)^{2}$$

Cancel out 3 and 3.

$$\left(\frac{2X^{2}Y^{-\frac{3}{4}}}{X^{\frac{3}{2}}}\right)^{2}$$

Cancel out X^{\frac{3}{2}} in both numerator and denominator.

$$\left(2Y^{-\frac{3}{4}}\sqrt{X}\right)^{2}$$

Expand \left(2Y^{-\frac{3}{4}}\sqrt{X}\right)^{2}.

$$2^{2}\left(Y^{-\frac{3}{4}}\right)^{2}\left(\sqrt{X}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply -\frac{3}{4} and 2 to get -\frac{3}{2}.

$$2^{2}Y^{-\frac{3}{2}}\left(\sqrt{X}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4Y^{-\frac{3}{2}}\left(\sqrt{X}\right)^{2}$$

Calculate \sqrt{X} to the power of 2 and get X.

$$4Y^{-\frac{3}{2}}X$$

Expand

$$\frac{4X}{Y^{\frac{3}{2}}}$$

Solution Steps

Divide 4X^{-1} by 2 to get 2X^{-1}.

$$\left(\frac{2X^{-1}X^{3}Y^{-\frac{3}{4}}}{\frac{\frac{6X^{\frac{3}{2}}Y^{-4}}{2}Y^{4}}{3}}\right)^{2}$$

To multiply powers of the same base, add their exponents. Add -1 and 3 to get 2.

$$\left(\frac{2X^{2}Y^{-\frac{3}{4}}}{\frac{\frac{6X^{\frac{3}{2}}Y^{-4}}{2}Y^{4}}{3}}\right)^{2}$$

Divide 6X^{\frac{3}{2}}Y^{-4} by 2 to get 3X^{\frac{3}{2}}Y^{-4}.

$$\left(\frac{2X^{2}Y^{-\frac{3}{4}}}{\frac{3X^{\frac{3}{2}}Y^{-4}Y^{4}}{3}}\right)^{2}$$

Multiply Y^{-4} and Y^{4} to get 1.

$$\left(\frac{2X^{2}Y^{-\frac{3}{4}}}{\frac{3X^{\frac{3}{2}}}{3}}\right)^{2}$$

Cancel out 3 and 3.

$$\left(\frac{2X^{2}Y^{-\frac{3}{4}}}{X^{\frac{3}{2}}}\right)^{2}$$

Cancel out X^{\frac{3}{2}} in both numerator and denominator.

$$\left(2Y^{-\frac{3}{4}}\sqrt{X}\right)^{2}$$

Expand \left(2Y^{-\frac{3}{4}}\sqrt{X}\right)^{2}.

$$2^{2}\left(Y^{-\frac{3}{4}}\right)^{2}\left(\sqrt{X}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply -\frac{3}{4} and 2 to get -\frac{3}{2}.

$$2^{2}Y^{-\frac{3}{2}}\left(\sqrt{X}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4Y^{-\frac{3}{2}}\left(\sqrt{X}\right)^{2}$$

Calculate \sqrt{X} to the power of 2 and get X.

$$4Y^{-\frac{3}{2}}X$$