Solve frac{64(x^{2}y^{4})^{1/2}}{(2x^{1/2}y^{2})^{2}}*frac{(8^{2}x^{6}y^{4})^frac{1{2}}}{(4x^3y^frac{1{2}})^2}

$$\frac { 64 ( x ^ { 2 } y ^ { 4 } ) ^ { \frac { 1 } { 2 } } } { ( 2 x ^ { \frac { 1 } { 2 } } y ^ { 2 } ) ^ { 2 } } \cdot \frac { ( 8 ^ { 2 } x ^ { 6 } y ^ { 4 } ) ^ { \frac { 1 } { 2 } } } { ( 4 x ^ { 3 } y ^ { \frac { 1 } { 2 } } ) ^ { 2 } }$$

Evaluate

$$\frac{8}{yx^{3}}$$

Solution Steps

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

$$\frac{64\left(x^{2}\right)^{\frac{1}{2}}\left(y^{4}\right)^{\frac{1}{2}}}{\left(2x^{\frac{1}{2}}y^{2}\right)^{2}}\times \frac{\left(8^{2}x^{6}y^{4}\right)^{\frac{1}{2}}}{\left(4x^{3}y^{\frac{1}{2}}\right)^{2}}$$

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

$$\frac{64x^{1}\left(y^{4}\right)^{\frac{1}{2}}}{\left(2x^{\frac{1}{2}}y^{2}\right)^{2}}\times \frac{\left(8^{2}x^{6}y^{4}\right)^{\frac{1}{2}}}{\left(4x^{3}y^{\frac{1}{2}}\right)^{2}}$$

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

$$\frac{64x^{1}y^{2}}{\left(2x^{\frac{1}{2}}y^{2}\right)^{2}}\times \frac{\left(8^{2}x^{6}y^{4}\right)^{\frac{1}{2}}}{\left(4x^{3}y^{\frac{1}{2}}\right)^{2}}$$

Calculate x to the power of 1 and get x.

$$\frac{64xy^{2}}{\left(2x^{\frac{1}{2}}y^{2}\right)^{2}}\times \frac{\left(8^{2}x^{6}y^{4}\right)^{\frac{1}{2}}}{\left(4x^{3}y^{\frac{1}{2}}\right)^{2}}$$

Expand \left(2x^{\frac{1}{2}}y^{2}\right)^{2}.

$$\frac{64xy^{2}}{2^{2}\left(x^{\frac{1}{2}}\right)^{2}\left(y^{2}\right)^{2}}\times \frac{\left(8^{2}x^{6}y^{4}\right)^{\frac{1}{2}}}{\left(4x^{3}y^{\frac{1}{2}}\right)^{2}}$$

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

$$\frac{64xy^{2}}{2^{2}x^{1}\left(y^{2}\right)^{2}}\times \frac{\left(8^{2}x^{6}y^{4}\right)^{\frac{1}{2}}}{\left(4x^{3}y^{\frac{1}{2}}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{64xy^{2}}{2^{2}x^{1}y^{4}}\times \frac{\left(8^{2}x^{6}y^{4}\right)^{\frac{1}{2}}}{\left(4x^{3}y^{\frac{1}{2}}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{64xy^{2}}{4x^{1}y^{4}}\times \frac{\left(8^{2}x^{6}y^{4}\right)^{\frac{1}{2}}}{\left(4x^{3}y^{\frac{1}{2}}\right)^{2}}$$

Calculate x to the power of 1 and get x.

$$\frac{64xy^{2}}{4xy^{4}}\times \frac{\left(8^{2}x^{6}y^{4}\right)^{\frac{1}{2}}}{\left(4x^{3}y^{\frac{1}{2}}\right)^{2}}$$

Cancel out 4xy^{2} in both numerator and denominator.

$$\frac{16}{y^{2}}\times \frac{\left(8^{2}x^{6}y^{4}\right)^{\frac{1}{2}}}{\left(4x^{3}y^{\frac{1}{2}}\right)^{2}}$$

Calculate 8 to the power of 2 and get 64.

$$\frac{16}{y^{2}}\times \frac{\left(64x^{6}y^{4}\right)^{\frac{1}{2}}}{\left(4x^{3}y^{\frac{1}{2}}\right)^{2}}$$

Expand \left(64x^{6}y^{4}\right)^{\frac{1}{2}}.

$$\frac{16}{y^{2}}\times \frac{64^{\frac{1}{2}}\left(x^{6}\right)^{\frac{1}{2}}\left(y^{4}\right)^{\frac{1}{2}}}{\left(4x^{3}y^{\frac{1}{2}}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and \frac{1}{2} to get 3.

$$\frac{16}{y^{2}}\times \frac{64^{\frac{1}{2}}x^{3}\left(y^{4}\right)^{\frac{1}{2}}}{\left(4x^{3}y^{\frac{1}{2}}\right)^{2}}$$

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

$$\frac{16}{y^{2}}\times \frac{64^{\frac{1}{2}}x^{3}y^{2}}{\left(4x^{3}y^{\frac{1}{2}}\right)^{2}}$$

Calculate 64 to the power of \frac{1}{2} and get 8.

$$\frac{16}{y^{2}}\times \frac{8x^{3}y^{2}}{\left(4x^{3}y^{\frac{1}{2}}\right)^{2}}$$

Expand \left(4x^{3}y^{\frac{1}{2}}\right)^{2}.

$$\frac{16}{y^{2}}\times \frac{8x^{3}y^{2}}{4^{2}\left(x^{3}\right)^{2}\left(y^{\frac{1}{2}}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{16}{y^{2}}\times \frac{8x^{3}y^{2}}{4^{2}x^{6}\left(y^{\frac{1}{2}}\right)^{2}}$$

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

$$\frac{16}{y^{2}}\times \frac{8x^{3}y^{2}}{4^{2}x^{6}y^{1}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{16}{y^{2}}\times \frac{8x^{3}y^{2}}{16x^{6}y^{1}}$$

Calculate y to the power of 1 and get y.

$$\frac{16}{y^{2}}\times \frac{8x^{3}y^{2}}{16x^{6}y}$$

Cancel out 8yx^{3} in both numerator and denominator.

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

Multiply \frac{16}{y^{2}} times \frac{y}{2x^{3}} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out 2y in both numerator and denominator.

$$\frac{8}{yx^{3}}$$

Differentiate w.r.t. x

$$-\frac{24}{yx^{4}}$$