Solve (a/b+b/a)^2-(a/b-b/a)^2

$$( \frac { a } { b } + \frac { b } { a } ) ^ { 2 } - ( \frac { a } { b } - \frac { b } { a } ) ^ { 2 }$$

Evaluate

$$4$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of b and a is ab. Multiply \frac{a}{b} times \frac{a}{a}. Multiply \frac{b}{a} times \frac{b}{b}.

$$\left(\frac{aa}{ab}+\frac{bb}{ab}\right)^{2}-\left(\frac{a}{b}-\frac{b}{a}\right)^{2}$$

Since \frac{aa}{ab} and \frac{bb}{ab} have the same denominator, add them by adding their numerators.

$$\left(\frac{aa+bb}{ab}\right)^{2}-\left(\frac{a}{b}-\frac{b}{a}\right)^{2}$$

Do the multiplications in aa+bb.

$$\left(\frac{a^{2}+b^{2}}{ab}\right)^{2}-\left(\frac{a}{b}-\frac{b}{a}\right)^{2}$$

To raise \frac{a^{2}+b^{2}}{ab} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(a^{2}+b^{2}\right)^{2}}{\left(ab\right)^{2}}-\left(\frac{a}{b}-\frac{b}{a}\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of b and a is ab. Multiply \frac{a}{b} times \frac{a}{a}. Multiply \frac{b}{a} times \frac{b}{b}.

$$\frac{\left(a^{2}+b^{2}\right)^{2}}{\left(ab\right)^{2}}-\left(\frac{aa}{ab}-\frac{bb}{ab}\right)^{2}$$

Since \frac{aa}{ab} and \frac{bb}{ab} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(a^{2}+b^{2}\right)^{2}}{\left(ab\right)^{2}}-\left(\frac{aa-bb}{ab}\right)^{2}$$

Do the multiplications in aa-bb.

$$\frac{\left(a^{2}+b^{2}\right)^{2}}{\left(ab\right)^{2}}-\left(\frac{a^{2}-b^{2}}{ab}\right)^{2}$$

To raise \frac{a^{2}-b^{2}}{ab} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(a^{2}+b^{2}\right)^{2}}{\left(ab\right)^{2}}-\frac{\left(a^{2}-b^{2}\right)^{2}}{\left(ab\right)^{2}}$$

Expand \left(ab\right)^{2}.

$$\frac{\left(a^{2}+b^{2}\right)^{2}}{\left(ab\right)^{2}}-\frac{\left(a^{2}-b^{2}\right)^{2}}{a^{2}b^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Expand \left(ab\right)^{2}.

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

Since \frac{\left(a^{2}+b^{2}\right)^{2}}{a^{2}b^{2}} and \frac{\left(a^{2}-b^{2}\right)^{2}}{a^{2}b^{2}} have the same denominator, subtract them by subtracting their numerators.

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

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

$$\frac{a^{4}+2a^{2}b^{2}+b^{4}-a^{4}+2a^{2}b^{2}-b^{4}}{a^{2}b^{2}}$$

Combine like terms in a^{4}+2a^{2}b^{2}+b^{4}-a^{4}+2a^{2}b^{2}-b^{4}.

$$\frac{4a^{2}b^{2}}{a^{2}b^{2}}$$

Cancel out a^{2}b^{2} in both numerator and denominator.

$$4$$

Factor

$$2^{2}$$