Solve (16x^{4}y^{5})^frac{1{4}}div(4x^frac{5{8}}y^frac{1{8}})^-2

$$( 16 x ^ { 4 } y ^ { 5 } ) ^ { \frac { 1 } { 4 } } \div ( 4 x ^ { \frac { 5 } { 8 } } y ^ { \frac { 1 } { 8 } } ) ^ { - 2 }$$

Evaluate

$$32y^{\frac{3}{2}}x^{\frac{9}{4}}$$

Solution Steps

Expand \left(16x^{4}y^{5}\right)^{\frac{1}{4}}.

$$\frac{16^{\frac{1}{4}}\left(x^{4}\right)^{\frac{1}{4}}\left(y^{5}\right)^{\frac{1}{4}}}{\left(4x^{\frac{5}{8}}y^{\frac{1}{8}}\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and \frac{1}{4} to get 1.

$$\frac{16^{\frac{1}{4}}x^{1}\left(y^{5}\right)^{\frac{1}{4}}}{\left(4x^{\frac{5}{8}}y^{\frac{1}{8}}\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and \frac{1}{4} to get \frac{5}{4}.

$$\frac{16^{\frac{1}{4}}x^{1}y^{\frac{5}{4}}}{\left(4x^{\frac{5}{8}}y^{\frac{1}{8}}\right)^{-2}}$$

Calculate 16 to the power of \frac{1}{4} and get 2.

$$\frac{2x^{1}y^{\frac{5}{4}}}{\left(4x^{\frac{5}{8}}y^{\frac{1}{8}}\right)^{-2}}$$

Calculate x to the power of 1 and get x.

$$\frac{2xy^{\frac{5}{4}}}{\left(4x^{\frac{5}{8}}y^{\frac{1}{8}}\right)^{-2}}$$

Expand \left(4x^{\frac{5}{8}}y^{\frac{1}{8}}\right)^{-2}.

$$\frac{2xy^{\frac{5}{4}}}{4^{-2}\left(x^{\frac{5}{8}}\right)^{-2}\left(y^{\frac{1}{8}}\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{5}{8} and -2 to get -\frac{5}{4}.

$$\frac{2xy^{\frac{5}{4}}}{4^{-2}x^{-\frac{5}{4}}\left(y^{\frac{1}{8}}\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{1}{8} and -2 to get -\frac{1}{4}.

$$\frac{2xy^{\frac{5}{4}}}{4^{-2}x^{-\frac{5}{4}}y^{-\frac{1}{4}}}$$

Calculate 4 to the power of -2 and get \frac{1}{16}.

$$\frac{2xy^{\frac{5}{4}}}{\frac{1}{16}x^{-\frac{5}{4}}y^{-\frac{1}{4}}}$$

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

$$\frac{2y^{\frac{3}{2}}x^{\frac{9}{4}}}{\frac{1}{16}}$$

Divide 2y^{\frac{3}{2}}x^{\frac{9}{4}} by \frac{1}{16} by multiplying 2y^{\frac{3}{2}}x^{\frac{9}{4}} by the reciprocal of \frac{1}{16}.

$$2y^{\frac{3}{2}}x^{\frac{9}{4}}\times 16$$

Multiply 2 and 16 to get 32.

$$32y^{\frac{3}{2}}x^{\frac{9}{4}}$$

Differentiate w.r.t. x

$$72x^{\frac{5}{4}}y^{\frac{3}{2}}$$