Solve 1/frac{1{x}}x^2-25/x^2-9*x+3/3-x/frac{1{3}(9-3x)x}

$$\frac{ 1 }{ \frac{ 1 }{ x } } \frac{ \frac{ { x }^{ 2 } -25 }{ { x }^{ 2 } -9 } \times \frac{ x+3 }{ 3-x } }{ \frac{ 1 }{ 3 } (9-3x)x }$$

Evaluate

$$-\frac{x^{2}-25}{\left(3-x\right)^{3}}$$

Short Solution Steps

Divide 1 by \frac{1}{x} by multiplying 1 by the reciprocal of \frac{1}{x}.

$$x\times \frac{\frac{x^{2}-25}{x^{2}-9}\times \frac{x+3}{3-x}}{\frac{1}{3}\left(9-3x\right)x}$$

Multiply \frac{x^{2}-25}{x^{2}-9} times \frac{x+3}{3-x} by multiplying numerator times numerator and denominator times denominator.

$$x\times \frac{\frac{\left(x^{2}-25\right)\left(x+3\right)}{\left(x^{2}-9\right)\left(3-x\right)}}{\frac{1}{3}\left(9-3x\right)x}$$

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

$$x\times \frac{\frac{\left(x-5\right)\left(x+3\right)\left(x+5\right)}{\left(x-3\right)\left(x+3\right)\left(-x+3\right)}}{\frac{1}{3}\left(9-3x\right)x}$$

Cancel out x+3 in both numerator and denominator.

$$x\times \frac{\frac{\left(x-5\right)\left(x+5\right)}{\left(x-3\right)\left(-x+3\right)}}{\frac{1}{3}\left(9-3x\right)x}$$

Express \frac{\frac{\left(x-5\right)\left(x+5\right)}{\left(x-3\right)\left(-x+3\right)}}{\frac{1}{3}\left(9-3x\right)x} as a single fraction.

$$x\times \frac{\left(x-5\right)\left(x+5\right)}{\left(x-3\right)\left(-x+3\right)\times \frac{1}{3}\left(9-3x\right)x}$$

Consider \left(x-5\right)\left(x+5\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 5.

$$x\times \frac{x^{2}-25}{\left(x-3\right)\left(-x+3\right)\times \frac{1}{3}\left(9-3x\right)x}$$

Use the distributive property to multiply x-3 by -x+3 and combine like terms.

$$x\times \frac{x^{2}-25}{\left(-x^{2}+6x-9\right)\times \frac{1}{3}\left(9-3x\right)x}$$

Use the distributive property to multiply -x^{2}+6x-9 by \frac{1}{3}.

$$x\times \frac{x^{2}-25}{\left(-\frac{1}{3}x^{2}+2x-3\right)\left(9-3x\right)x}$$

Use the distributive property to multiply -\frac{1}{3}x^{2}+2x-3 by 9-3x and combine like terms.

$$x\times \frac{x^{2}-25}{\left(-9x^{2}+x^{3}+27x-27\right)x}$$

Use the distributive property to multiply -9x^{2}+x^{3}+27x-27 by x.

$$x\times \frac{x^{2}-25}{-9x^{3}+x^{4}+27x^{2}-27x}$$

Express x\times \frac{x^{2}-25}{-9x^{3}+x^{4}+27x^{2}-27x} as a single fraction.

$$\frac{x\left(x^{2}-25\right)}{-9x^{3}+x^{4}+27x^{2}-27x}$$

Factor the expressions that are not already factored.

$$\frac{x\left(x-5\right)\left(x+5\right)}{x\left(x-3\right)^{3}}$$

Cancel out x in both numerator and denominator.

$$\frac{\left(x-5\right)\left(x+5\right)}{\left(x-3\right)^{3}}$$

Expand the expression.

$$\frac{x^{2}-25}{x^{3}-9x^{2}+27x-27}$$

Expand

$$-\frac{x^{2}-25}{\left(3-x\right)^{3}}$$

Short Solution Steps

Divide 1 by \frac{1}{x} by multiplying 1 by the reciprocal of \frac{1}{x}.

$$x\times \frac{\frac{x^{2}-25}{x^{2}-9}\times \frac{x+3}{3-x}}{\frac{1}{3}\left(9-3x\right)x}$$

Multiply \frac{x^{2}-25}{x^{2}-9} times \frac{x+3}{3-x} by multiplying numerator times numerator and denominator times denominator.

$$x\times \frac{\frac{\left(x^{2}-25\right)\left(x+3\right)}{\left(x^{2}-9\right)\left(3-x\right)}}{\frac{1}{3}\left(9-3x\right)x}$$

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

$$x\times \frac{\frac{\left(x-5\right)\left(x+3\right)\left(x+5\right)}{\left(x-3\right)\left(x+3\right)\left(-x+3\right)}}{\frac{1}{3}\left(9-3x\right)x}$$

Cancel out x+3 in both numerator and denominator.

$$x\times \frac{\frac{\left(x-5\right)\left(x+5\right)}{\left(x-3\right)\left(-x+3\right)}}{\frac{1}{3}\left(9-3x\right)x}$$

Express \frac{\frac{\left(x-5\right)\left(x+5\right)}{\left(x-3\right)\left(-x+3\right)}}{\frac{1}{3}\left(9-3x\right)x} as a single fraction.

$$x\times \frac{\left(x-5\right)\left(x+5\right)}{\left(x-3\right)\left(-x+3\right)\times \frac{1}{3}\left(9-3x\right)x}$$

Consider \left(x-5\right)\left(x+5\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 5.

$$x\times \frac{x^{2}-25}{\left(x-3\right)\left(-x+3\right)\times \frac{1}{3}\left(9-3x\right)x}$$

Use the distributive property to multiply x-3 by -x+3 and combine like terms.

$$x\times \frac{x^{2}-25}{\left(-x^{2}+6x-9\right)\times \frac{1}{3}\left(9-3x\right)x}$$

Use the distributive property to multiply -x^{2}+6x-9 by \frac{1}{3}.

$$x\times \frac{x^{2}-25}{\left(-\frac{1}{3}x^{2}+2x-3\right)\left(9-3x\right)x}$$

Use the distributive property to multiply -\frac{1}{3}x^{2}+2x-3 by 9-3x and combine like terms.

$$x\times \frac{x^{2}-25}{\left(-9x^{2}+x^{3}+27x-27\right)x}$$

Use the distributive property to multiply -9x^{2}+x^{3}+27x-27 by x.

$$x\times \frac{x^{2}-25}{-9x^{3}+x^{4}+27x^{2}-27x}$$

Express x\times \frac{x^{2}-25}{-9x^{3}+x^{4}+27x^{2}-27x} as a single fraction.

$$\frac{x\left(x^{2}-25\right)}{-9x^{3}+x^{4}+27x^{2}-27x}$$

Factor the expressions that are not already factored.

$$\frac{x\left(x-5\right)\left(x+5\right)}{x\left(x-3\right)^{3}}$$

Cancel out x in both numerator and denominator.

$$\frac{\left(x-5\right)\left(x+5\right)}{\left(x-3\right)^{3}}$$

Expand the expression.

$$\frac{x^{2}-25}{x^{3}-9x^{2}+27x-27}$$