Solve 1+a-x/a+x/1-frac{a-x{a+x}}divfrac{1+frac{a^{2}-x^{2}}{a^{2}+x^{2}}}{1-a^2-x^2/a^2+x^2}

$$\frac { 1 + \frac { a - x } { a + x } } { 1 - \frac { a - x } { a + x } } \div \frac { 1 + \frac { a ^ { 2 } - x ^ { 2 } } { a ^ { 2 } + x ^ { 2 } } } { 1 - \frac { a ^ { 2 } - x ^ { 2 } } { a ^ { 2 } + x ^ { 2 } } }$$

Evaluate

$$\frac{x}{a}$$

Short Solution Steps

Divide \frac{1+\frac{a-x}{a+x}}{1-\frac{a-x}{a+x}} by \frac{1+\frac{a^{2}-x^{2}}{a^{2}+x^{2}}}{1-\frac{a^{2}-x^{2}}{a^{2}+x^{2}}} by multiplying \frac{1+\frac{a-x}{a+x}}{1-\frac{a-x}{a+x}} by the reciprocal of \frac{1+\frac{a^{2}-x^{2}}{a^{2}+x^{2}}}{1-\frac{a^{2}-x^{2}}{a^{2}+x^{2}}}.

$$\frac{\left(1+\frac{a-x}{a+x}\right)\left(1-\frac{a^{2}-x^{2}}{a^{2}+x^{2}}\right)}{\left(1-\frac{a-x}{a+x}\right)\left(1+\frac{a^{2}-x^{2}}{a^{2}+x^{2}}\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{a+x}{a+x}.

$$\frac{\left(\frac{a+x}{a+x}+\frac{a-x}{a+x}\right)\left(1-\frac{a^{2}-x^{2}}{a^{2}+x^{2}}\right)}{\left(1-\frac{a-x}{a+x}\right)\left(1+\frac{a^{2}-x^{2}}{a^{2}+x^{2}}\right)}$$

Since \frac{a+x}{a+x} and \frac{a-x}{a+x} have the same denominator, add them by adding their numerators.

$$\frac{\frac{a+x+a-x}{a+x}\left(1-\frac{a^{2}-x^{2}}{a^{2}+x^{2}}\right)}{\left(1-\frac{a-x}{a+x}\right)\left(1+\frac{a^{2}-x^{2}}{a^{2}+x^{2}}\right)}$$

Combine like terms in a+x+a-x.

$$\frac{\frac{2a}{a+x}\left(1-\frac{a^{2}-x^{2}}{a^{2}+x^{2}}\right)}{\left(1-\frac{a-x}{a+x}\right)\left(1+\frac{a^{2}-x^{2}}{a^{2}+x^{2}}\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{a^{2}+x^{2}}{a^{2}+x^{2}}.

$$\frac{\frac{2a}{a+x}\left(\frac{a^{2}+x^{2}}{a^{2}+x^{2}}-\frac{a^{2}-x^{2}}{a^{2}+x^{2}}\right)}{\left(1-\frac{a-x}{a+x}\right)\left(1+\frac{a^{2}-x^{2}}{a^{2}+x^{2}}\right)}$$

Since \frac{a^{2}+x^{2}}{a^{2}+x^{2}} and \frac{a^{2}-x^{2}}{a^{2}+x^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in a^{2}+x^{2}-\left(a^{2}-x^{2}\right).

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

Combine like terms in a^{2}+x^{2}-a^{2}+x^{2}.

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

Multiply \frac{2a}{a+x} times \frac{2x^{2}}{a^{2}+x^{2}} by multiplying numerator times numerator and denominator times denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{a+x}{a+x}.

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

Since \frac{a+x}{a+x} and \frac{a-x}{a+x} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in a+x-\left(a-x\right).

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

Combine like terms in a+x-a+x.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{a^{2}+x^{2}}{a^{2}+x^{2}}.

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

Since \frac{a^{2}+x^{2}}{a^{2}+x^{2}} and \frac{a^{2}-x^{2}}{a^{2}+x^{2}} have the same denominator, add them by adding their numerators.

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

Combine like terms in a^{2}+x^{2}+a^{2}-x^{2}.

$$\frac{\frac{2a\times 2x^{2}}{\left(a+x\right)\left(a^{2}+x^{2}\right)}}{\frac{2x}{a+x}\times \frac{2a^{2}}{a^{2}+x^{2}}}$$

Multiply \frac{2x}{a+x} times \frac{2a^{2}}{a^{2}+x^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{2a\times 2x^{2}}{\left(a+x\right)\left(a^{2}+x^{2}\right)}}{\frac{2x\times 2a^{2}}{\left(a+x\right)\left(a^{2}+x^{2}\right)}}$$

Divide \frac{2a\times 2x^{2}}{\left(a+x\right)\left(a^{2}+x^{2}\right)} by \frac{2x\times 2a^{2}}{\left(a+x\right)\left(a^{2}+x^{2}\right)} by multiplying \frac{2a\times 2x^{2}}{\left(a+x\right)\left(a^{2}+x^{2}\right)} by the reciprocal of \frac{2x\times 2a^{2}}{\left(a+x\right)\left(a^{2}+x^{2}\right)}.

$$\frac{2a\times 2x^{2}\left(a+x\right)\left(a^{2}+x^{2}\right)}{\left(a+x\right)\left(a^{2}+x^{2}\right)\times 2x\times 2a^{2}}$$

Cancel out 2\times 2ax\left(x+a\right)\left(x^{2}+a^{2}\right) in both numerator and denominator.

$$\frac{x}{a}$$

Expand

$$\frac{x}{a}$$

Short Solution Steps

Divide \frac{1+\frac{a-x}{a+x}}{1-\frac{a-x}{a+x}} by \frac{1+\frac{a^{2}-x^{2}}{a^{2}+x^{2}}}{1-\frac{a^{2}-x^{2}}{a^{2}+x^{2}}} by multiplying \frac{1+\frac{a-x}{a+x}}{1-\frac{a-x}{a+x}} by the reciprocal of \frac{1+\frac{a^{2}-x^{2}}{a^{2}+x^{2}}}{1-\frac{a^{2}-x^{2}}{a^{2}+x^{2}}}.

$$\frac{\left(1+\frac{a-x}{a+x}\right)\left(1-\frac{a^{2}-x^{2}}{a^{2}+x^{2}}\right)}{\left(1-\frac{a-x}{a+x}\right)\left(1+\frac{a^{2}-x^{2}}{a^{2}+x^{2}}\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{a+x}{a+x}.

$$\frac{\left(\frac{a+x}{a+x}+\frac{a-x}{a+x}\right)\left(1-\frac{a^{2}-x^{2}}{a^{2}+x^{2}}\right)}{\left(1-\frac{a-x}{a+x}\right)\left(1+\frac{a^{2}-x^{2}}{a^{2}+x^{2}}\right)}$$

Since \frac{a+x}{a+x} and \frac{a-x}{a+x} have the same denominator, add them by adding their numerators.

$$\frac{\frac{a+x+a-x}{a+x}\left(1-\frac{a^{2}-x^{2}}{a^{2}+x^{2}}\right)}{\left(1-\frac{a-x}{a+x}\right)\left(1+\frac{a^{2}-x^{2}}{a^{2}+x^{2}}\right)}$$

Combine like terms in a+x+a-x.

$$\frac{\frac{2a}{a+x}\left(1-\frac{a^{2}-x^{2}}{a^{2}+x^{2}}\right)}{\left(1-\frac{a-x}{a+x}\right)\left(1+\frac{a^{2}-x^{2}}{a^{2}+x^{2}}\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{a^{2}+x^{2}}{a^{2}+x^{2}}.

$$\frac{\frac{2a}{a+x}\left(\frac{a^{2}+x^{2}}{a^{2}+x^{2}}-\frac{a^{2}-x^{2}}{a^{2}+x^{2}}\right)}{\left(1-\frac{a-x}{a+x}\right)\left(1+\frac{a^{2}-x^{2}}{a^{2}+x^{2}}\right)}$$

Since \frac{a^{2}+x^{2}}{a^{2}+x^{2}} and \frac{a^{2}-x^{2}}{a^{2}+x^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in a^{2}+x^{2}-\left(a^{2}-x^{2}\right).

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

Combine like terms in a^{2}+x^{2}-a^{2}+x^{2}.

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

Multiply \frac{2a}{a+x} times \frac{2x^{2}}{a^{2}+x^{2}} by multiplying numerator times numerator and denominator times denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{a+x}{a+x}.

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

Since \frac{a+x}{a+x} and \frac{a-x}{a+x} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in a+x-\left(a-x\right).

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

Combine like terms in a+x-a+x.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{a^{2}+x^{2}}{a^{2}+x^{2}}.

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

Since \frac{a^{2}+x^{2}}{a^{2}+x^{2}} and \frac{a^{2}-x^{2}}{a^{2}+x^{2}} have the same denominator, add them by adding their numerators.

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

Combine like terms in a^{2}+x^{2}+a^{2}-x^{2}.

$$\frac{\frac{2a\times 2x^{2}}{\left(a+x\right)\left(a^{2}+x^{2}\right)}}{\frac{2x}{a+x}\times \frac{2a^{2}}{a^{2}+x^{2}}}$$

Multiply \frac{2x}{a+x} times \frac{2a^{2}}{a^{2}+x^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{2a\times 2x^{2}}{\left(a+x\right)\left(a^{2}+x^{2}\right)}}{\frac{2x\times 2a^{2}}{\left(a+x\right)\left(a^{2}+x^{2}\right)}}$$

Divide \frac{2a\times 2x^{2}}{\left(a+x\right)\left(a^{2}+x^{2}\right)} by \frac{2x\times 2a^{2}}{\left(a+x\right)\left(a^{2}+x^{2}\right)} by multiplying \frac{2a\times 2x^{2}}{\left(a+x\right)\left(a^{2}+x^{2}\right)} by the reciprocal of \frac{2x\times 2a^{2}}{\left(a+x\right)\left(a^{2}+x^{2}\right)}.

$$\frac{2a\times 2x^{2}\left(a+x\right)\left(a^{2}+x^{2}\right)}{\left(a+x\right)\left(a^{2}+x^{2}\right)\times 2x\times 2a^{2}}$$

Cancel out 2\times 2ax\left(x+a\right)\left(x^{2}+a^{2}\right) in both numerator and denominator.

$$\frac{x}{a}$$