Solve (frac{54x^{2}y^{2}z^{7}}{16x^-1y^5z^4})^frac{1{3}}

$$( \frac { 54 x ^ { 2 } y ^ { 2 } z ^ { 7 } } { 16 x ^ { - 1 } y ^ { 5 } z ^ { 4 } } ) ^ { \frac { 1 } { 3 } }$$

Evaluate

$$\frac{3xz}{2y}$$

Solution Steps

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

$$\left(\frac{27x^{2}z^{3}}{8\times \frac{1}{x}y^{3}}\right)^{\frac{1}{3}}$$

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

$$\left(\frac{27x^{3}z^{3}}{8y^{3}}\right)^{\frac{1}{3}}$$

To raise \frac{27x^{3}z^{3}}{8y^{3}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(27x^{3}z^{3}\right)^{\frac{1}{3}}}{\left(8y^{3}\right)^{\frac{1}{3}}}$$

Expand \left(27x^{3}z^{3}\right)^{\frac{1}{3}}.

$$\frac{27^{\frac{1}{3}}\left(x^{3}\right)^{\frac{1}{3}}\left(z^{3}\right)^{\frac{1}{3}}}{\left(8y^{3}\right)^{\frac{1}{3}}}$$

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

$$\frac{27^{\frac{1}{3}}x^{1}\left(z^{3}\right)^{\frac{1}{3}}}{\left(8y^{3}\right)^{\frac{1}{3}}}$$

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

$$\frac{27^{\frac{1}{3}}x^{1}z^{1}}{\left(8y^{3}\right)^{\frac{1}{3}}}$$

Calculate 27 to the power of \frac{1}{3} and get 3.

$$\frac{3x^{1}z^{1}}{\left(8y^{3}\right)^{\frac{1}{3}}}$$

Calculate x to the power of 1 and get x.

$$\frac{3xz^{1}}{\left(8y^{3}\right)^{\frac{1}{3}}}$$

Calculate z to the power of 1 and get z.

$$\frac{3xz}{\left(8y^{3}\right)^{\frac{1}{3}}}$$

Expand \left(8y^{3}\right)^{\frac{1}{3}}.

$$\frac{3xz}{8^{\frac{1}{3}}\left(y^{3}\right)^{\frac{1}{3}}}$$

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

$$\frac{3xz}{8^{\frac{1}{3}}y^{1}}$$

Calculate 8 to the power of \frac{1}{3} and get 2.

$$\frac{3xz}{2y^{1}}$$

Calculate y to the power of 1 and get y.

$$\frac{3xz}{2y}$$

Differentiate w.r.t. x

$$\frac{3z}{2y}$$