Solve frac{6x^{2}}{7y^{2}}div2/49xydivx^3/x^2y+xy^2+y^3

$$\frac { 6 x ^ { 2 } } { 7 y ^ { 2 } } \div \frac { 2 } { 49 x y } \div \frac { x ^ { 3 } } { x ^ { 2 } y + x y ^ { 2 } + y ^ { 3 } }$$

Evaluate

$$21\left(x^{2}+xy+y^{2}\right)$$

Short Solution Steps

Divide \frac{\frac{6x^{2}}{7y^{2}}}{\frac{2}{49xy}} by \frac{x^{3}}{x^{2}y+xy^{2}+y^{3}} by multiplying \frac{\frac{6x^{2}}{7y^{2}}}{\frac{2}{49xy}} by the reciprocal of \frac{x^{3}}{x^{2}y+xy^{2}+y^{3}}.

$$\frac{\frac{6x^{2}}{7y^{2}}\left(x^{2}y+xy^{2}+y^{3}\right)}{\frac{2}{49xy}x^{3}}$$

Express \frac{6x^{2}}{7y^{2}}\left(x^{2}y+xy^{2}+y^{3}\right) as a single fraction.

$$\frac{\frac{6x^{2}\left(x^{2}y+xy^{2}+y^{3}\right)}{7y^{2}}}{\frac{2}{49xy}x^{3}}$$

Express \frac{2}{49xy}x^{3} as a single fraction.

$$\frac{\frac{6x^{2}\left(x^{2}y+xy^{2}+y^{3}\right)}{7y^{2}}}{\frac{2x^{3}}{49xy}}$$

Cancel out x in both numerator and denominator.

$$\frac{\frac{6x^{2}\left(x^{2}y+xy^{2}+y^{3}\right)}{7y^{2}}}{\frac{2x^{2}}{49y}}$$

Divide \frac{6x^{2}\left(x^{2}y+xy^{2}+y^{3}\right)}{7y^{2}} by \frac{2x^{2}}{49y} by multiplying \frac{6x^{2}\left(x^{2}y+xy^{2}+y^{3}\right)}{7y^{2}} by the reciprocal of \frac{2x^{2}}{49y}.

$$\frac{6x^{2}\left(x^{2}y+xy^{2}+y^{3}\right)\times 49y}{7y^{2}\times 2x^{2}}$$

Cancel out 2\times 7yx^{2} in both numerator and denominator.

$$\frac{3\times 7\left(xy^{2}+y^{3}+yx^{2}\right)}{y}$$

Factor the expressions that are not already factored.

$$\frac{3\times 7y\left(x^{2}+xy+y^{2}\right)}{y}$$

Cancel out y in both numerator and denominator.

$$3\times 7\left(x^{2}+xy+y^{2}\right)$$

Expand the expression.

$$21x^{2}+21xy+21y^{2}$$

Differentiate w.r.t. x

$$21\left(2x+y\right)$$