Solve x+6/left(x+7right)left(x-7right)*x-7/left(x+6right)left(x-6right)

$$\frac{ x+6 }{ \left( x+7 \right) \left( x-7 \right) } \times \frac{ x-7 }{ \left( x+6 \right) \left( x-6 \right) }$$

Evaluate

$$\frac{1}{\left(x-6\right)\left(x+7\right)}$$

Short Solution Steps

Consider \left(x+7\right)\left(x-7\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{x+6}{x^{2}-7^{2}}\times \frac{x-7}{\left(x+6\right)\left(x-6\right)}$$

Calculate 7 to the power of 2 and get 49.

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

Consider \left(x+6\right)\left(x-6\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{x+6}{x^{2}-49}\times \frac{x-7}{x^{2}-6^{2}}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{x+6}{x^{2}-49}\times \frac{x-7}{x^{2}-36}$$

Multiply \frac{x+6}{x^{2}-49} times \frac{x-7}{x^{2}-36} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x+6\right)\left(x-7\right)}{\left(x^{2}-49\right)\left(x^{2}-36\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(x-7\right)\left(x+6\right)}{\left(x-7\right)\left(x-6\right)\left(x+6\right)\left(x+7\right)}$$

Cancel out \left(x-7\right)\left(x+6\right) in both numerator and denominator.

$$\frac{1}{\left(x-6\right)\left(x+7\right)}$$

Expand the expression.

$$\frac{1}{x^{2}+x-42}$$

Expand

$$\frac{1}{\left(x-6\right)\left(x+7\right)}$$

Short Solution Steps

Consider \left(x+7\right)\left(x-7\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{x+6}{x^{2}-7^{2}}\times \frac{x-7}{\left(x+6\right)\left(x-6\right)}$$

Calculate 7 to the power of 2 and get 49.

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

Consider \left(x+6\right)\left(x-6\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{x+6}{x^{2}-49}\times \frac{x-7}{x^{2}-6^{2}}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{x+6}{x^{2}-49}\times \frac{x-7}{x^{2}-36}$$

Multiply \frac{x+6}{x^{2}-49} times \frac{x-7}{x^{2}-36} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x+6\right)\left(x-7\right)}{\left(x^{2}-49\right)\left(x^{2}-36\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(x-7\right)\left(x+6\right)}{\left(x-7\right)\left(x-6\right)\left(x+6\right)\left(x+7\right)}$$

Cancel out \left(x-7\right)\left(x+6\right) in both numerator and denominator.

$$\frac{1}{\left(x-6\right)\left(x+7\right)}$$

Expand the expression.

$$\frac{1}{x^{2}+x-42}$$