Solve (frac{4X^{-1/2}X^{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

$$\frac{16X^{2}Y^{\frac{23}{6}}}{9}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -\frac{1}{2} and 3 to get \frac{5}{2}.

$$\left(\frac{4X^{\frac{5}{2}}Y^{-\frac{3}{4}}}{\frac{6X^{\frac{3}{2}}Y^{-4}}{2}Y^{\frac{4}{3}}}\right)^{2}$$

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

$$\left(\frac{4X^{\frac{5}{2}}Y^{-\frac{3}{4}}}{3X^{\frac{3}{2}}Y^{-4}Y^{\frac{4}{3}}}\right)^{2}$$

To multiply powers of the same base, add their exponents. Add -4 and \frac{4}{3} to get -\frac{8}{3}.

$$\left(\frac{4X^{\frac{5}{2}}Y^{-\frac{3}{4}}}{3X^{\frac{3}{2}}Y^{-\frac{8}{3}}}\right)^{2}$$

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

$$\left(\frac{4Y^{-\frac{3}{4}}X}{3Y^{-\frac{8}{3}}}\right)^{2}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\left(\frac{4XY^{\frac{23}{12}}}{3}\right)^{2}$$

To raise \frac{4XY^{\frac{23}{12}}}{3} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(4XY^{\frac{23}{12}}\right)^{2}}{3^{2}}$$

Expand \left(4XY^{\frac{23}{12}}\right)^{2}.

$$\frac{4^{2}X^{2}\left(Y^{\frac{23}{12}}\right)^{2}}{3^{2}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{23}{12} and 2 to get \frac{23}{6}.

$$\frac{4^{2}X^{2}Y^{\frac{23}{6}}}{3^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{16X^{2}Y^{\frac{23}{6}}}{3^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{16X^{2}Y^{\frac{23}{6}}}{9}$$

Expand

$$\frac{16X^{2}Y^{\frac{23}{6}}}{9}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -\frac{1}{2} and 3 to get \frac{5}{2}.

$$\left(\frac{4X^{\frac{5}{2}}Y^{-\frac{3}{4}}}{\frac{6X^{\frac{3}{2}}Y^{-4}}{2}Y^{\frac{4}{3}}}\right)^{2}$$

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

$$\left(\frac{4X^{\frac{5}{2}}Y^{-\frac{3}{4}}}{3X^{\frac{3}{2}}Y^{-4}Y^{\frac{4}{3}}}\right)^{2}$$

To multiply powers of the same base, add their exponents. Add -4 and \frac{4}{3} to get -\frac{8}{3}.

$$\left(\frac{4X^{\frac{5}{2}}Y^{-\frac{3}{4}}}{3X^{\frac{3}{2}}Y^{-\frac{8}{3}}}\right)^{2}$$

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

$$\left(\frac{4Y^{-\frac{3}{4}}X}{3Y^{-\frac{8}{3}}}\right)^{2}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\left(\frac{4XY^{\frac{23}{12}}}{3}\right)^{2}$$

To raise \frac{4XY^{\frac{23}{12}}}{3} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(4XY^{\frac{23}{12}}\right)^{2}}{3^{2}}$$

Expand \left(4XY^{\frac{23}{12}}\right)^{2}.

$$\frac{4^{2}X^{2}\left(Y^{\frac{23}{12}}\right)^{2}}{3^{2}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{23}{12} and 2 to get \frac{23}{6}.

$$\frac{4^{2}X^{2}Y^{\frac{23}{6}}}{3^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{16X^{2}Y^{\frac{23}{6}}}{3^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{16X^{2}Y^{\frac{23}{6}}}{9}$$