Solve frac{9x^{2}-25y^{2}}{27x^{3}-125y^{3}}div6x+10y/5x-25y*9x^2+15xy+25y^2/9x^2-18xy+5y^2=

$$\frac { 9 x ^ { 2 } - 25 y ^ { 2 } } { 27 x ^ { 3 } - 125 y ^ { 3 } } \div \frac { 6 x + 10 y } { 5 x - 25 y } \cdot \frac { 9 x ^ { 2 } + 15 x y + 25 y ^ { 2 } } { 9 x ^ { 2 } - 18 x y + 5 y ^ { 2 } } =$$

Evaluate

$$\frac{5\left(x-5y\right)}{2\left(3x-y\right)\left(3x-5y\right)}$$

Short Solution Steps

Divide \frac{9x^{2}-25y^{2}}{27x^{3}-125y^{3}} by \frac{6x+10y}{5x-25y} by multiplying \frac{9x^{2}-25y^{2}}{27x^{3}-125y^{3}} by the reciprocal of \frac{6x+10y}{5x-25y}.

$$\frac{\left(9x^{2}-25y^{2}\right)\left(5x-25y\right)}{\left(27x^{3}-125y^{3}\right)\left(6x+10y\right)}\times \frac{9x^{2}+15xy+25y^{2}}{9x^{2}-18xy+5y^{2}}$$

Factor the expressions that are not already factored in \frac{\left(9x^{2}-25y^{2}\right)\left(5x-25y\right)}{\left(27x^{3}-125y^{3}\right)\left(6x+10y\right)}.

$$\frac{5\left(x-5y\right)\left(3x-5y\right)\left(3x+5y\right)}{2\left(3x-5y\right)\left(3x+5y\right)\left(9x^{2}+15xy+25y^{2}\right)}\times \frac{9x^{2}+15xy+25y^{2}}{9x^{2}-18xy+5y^{2}}$$

Cancel out \left(3x-5y\right)\left(3x+5y\right) in both numerator and denominator.

$$\frac{5\left(x-5y\right)}{2\left(9x^{2}+15xy+25y^{2}\right)}\times \frac{9x^{2}+15xy+25y^{2}}{9x^{2}-18xy+5y^{2}}$$

Multiply \frac{5\left(x-5y\right)}{2\left(9x^{2}+15xy+25y^{2}\right)} times \frac{9x^{2}+15xy+25y^{2}}{9x^{2}-18xy+5y^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{5\left(x-5y\right)\left(9x^{2}+15xy+25y^{2}\right)}{2\left(9x^{2}+15xy+25y^{2}\right)\left(9x^{2}-18xy+5y^{2}\right)}$$

Cancel out 9x^{2}+15xy+25y^{2} in both numerator and denominator.

$$\frac{5\left(x-5y\right)}{2\left(9x^{2}-18xy+5y^{2}\right)}$$

Use the distributive property to multiply 5 by x-5y.

$$\frac{5x-25y}{2\left(9x^{2}-18xy+5y^{2}\right)}$$

Use the distributive property to multiply 2 by 9x^{2}-18xy+5y^{2}.

$$\frac{5x-25y}{18x^{2}-36xy+10y^{2}}$$

Expand

$$-\frac{5\left(5y-x\right)}{2\left(y-3x\right)\left(5y-3x\right)}$$

Short Solution Steps

Divide \frac{9x^{2}-25y^{2}}{27x^{3}-125y^{3}} by \frac{6x+10y}{5x-25y} by multiplying \frac{9x^{2}-25y^{2}}{27x^{3}-125y^{3}} by the reciprocal of \frac{6x+10y}{5x-25y}.

$$\frac{\left(9x^{2}-25y^{2}\right)\left(5x-25y\right)}{\left(27x^{3}-125y^{3}\right)\left(6x+10y\right)}\times \frac{9x^{2}+15xy+25y^{2}}{9x^{2}-18xy+5y^{2}}$$

Factor the expressions that are not already factored in \frac{\left(9x^{2}-25y^{2}\right)\left(5x-25y\right)}{\left(27x^{3}-125y^{3}\right)\left(6x+10y\right)}.

$$\frac{5\left(x-5y\right)\left(3x-5y\right)\left(3x+5y\right)}{2\left(3x-5y\right)\left(3x+5y\right)\left(9x^{2}+15xy+25y^{2}\right)}\times \frac{9x^{2}+15xy+25y^{2}}{9x^{2}-18xy+5y^{2}}$$

Cancel out \left(3x-5y\right)\left(3x+5y\right) in both numerator and denominator.

$$\frac{5\left(x-5y\right)}{2\left(9x^{2}+15xy+25y^{2}\right)}\times \frac{9x^{2}+15xy+25y^{2}}{9x^{2}-18xy+5y^{2}}$$

Multiply \frac{5\left(x-5y\right)}{2\left(9x^{2}+15xy+25y^{2}\right)} times \frac{9x^{2}+15xy+25y^{2}}{9x^{2}-18xy+5y^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{5\left(x-5y\right)\left(9x^{2}+15xy+25y^{2}\right)}{2\left(9x^{2}+15xy+25y^{2}\right)\left(9x^{2}-18xy+5y^{2}\right)}$$

Cancel out 9x^{2}+15xy+25y^{2} in both numerator and denominator.

$$\frac{5\left(x-5y\right)}{2\left(9x^{2}-18xy+5y^{2}\right)}$$

Use the distributive property to multiply 5 by x-5y.

$$\frac{5x-25y}{2\left(9x^{2}-18xy+5y^{2}\right)}$$

Use the distributive property to multiply 2 by 9x^{2}-18xy+5y^{2}.

$$\frac{5x-25y}{18x^{2}-36xy+10y^{2}}$$