Solve (frac{36x^{4}y^3}{4x^8y^-1})^frac{1{2}}

$$( \frac { 36 x ^ { 4 } y ^ { 3 } } { 4 x ^ { 8 } y ^ { - 1 } } ) ^ { \frac { 1 } { 2 } }$$

Evaluate

$$3\times \left(\frac{y}{x}\right)^{2}$$

Solution Steps

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

$$\left(\frac{9y^{3}}{\frac{1}{y}x^{4}}\right)^{\frac{1}{2}}$$

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

$$\left(\frac{9y^{4}}{x^{4}}\right)^{\frac{1}{2}}$$

To raise \frac{9y^{4}}{x^{4}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(9y^{4}\right)^{\frac{1}{2}}}{\left(x^{4}\right)^{\frac{1}{2}}}$$

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

$$\frac{\left(9y^{4}\right)^{\frac{1}{2}}}{x^{2}}$$

Expand \left(9y^{4}\right)^{\frac{1}{2}}.

$$\frac{9^{\frac{1}{2}}\left(y^{4}\right)^{\frac{1}{2}}}{x^{2}}$$

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

$$\frac{9^{\frac{1}{2}}y^{2}}{x^{2}}$$

Calculate 9 to the power of \frac{1}{2} and get 3.

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

Differentiate w.r.t. x

$$-\frac{6y^{2}}{x^{3}}$$