Solve {left(frac{1/3y^2right)}^3}{{left(1/3x^2yright)}^4}

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

Evaluate

$$\frac{3y^{2}}{x^{8}}$$

Solution Steps

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

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

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

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

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

$$\frac{\frac{1}{27}y^{6}}{\left(\frac{1}{3}x^{2}y\right)^{4}}$$

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

$$\frac{\frac{1}{27}y^{6}}{\left(\frac{1}{3}\right)^{4}\left(x^{2}\right)^{4}y^{4}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 4 to get 8.

$$\frac{\frac{1}{27}y^{6}}{\left(\frac{1}{3}\right)^{4}x^{8}y^{4}}$$

Calculate \frac{1}{3} to the power of 4 and get \frac{1}{81}.

$$\frac{\frac{1}{27}y^{6}}{\frac{1}{81}x^{8}y^{4}}$$

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

$$\frac{\frac{1}{27}y^{2}}{\frac{1}{81}x^{8}}$$

Expand

$$\frac{3y^{2}}{x^{8}}$$

Solution Steps

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

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

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

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

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

$$\frac{\frac{1}{27}y^{6}}{\left(\frac{1}{3}x^{2}y\right)^{4}}$$

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

$$\frac{\frac{1}{27}y^{6}}{\left(\frac{1}{3}\right)^{4}\left(x^{2}\right)^{4}y^{4}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 4 to get 8.

$$\frac{\frac{1}{27}y^{6}}{\left(\frac{1}{3}\right)^{4}x^{8}y^{4}}$$

Calculate \frac{1}{3} to the power of 4 and get \frac{1}{81}.

$$\frac{\frac{1}{27}y^{6}}{\frac{1}{81}x^{8}y^{4}}$$

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

$$\frac{\frac{1}{27}y^{2}}{\frac{1}{81}x^{8}}$$