Solve frac{(sqrt{7})^2-(sqrt{3})^2}{4(sqrt{7}+sqrt{3})}

$$\frac { ( \sqrt { 7 } ) ^ { 2 } - ( \sqrt { 3 } ) ^ { 2 } } { 4 ( \sqrt { 7 } + \sqrt { 3 } ) }$$

Evaluate

$$\frac{\sqrt{7}-\sqrt{3}}{4}\approx 0.228425126$$

Solution Steps

The square of \sqrt{7} is 7.

$$\frac{7-\left(\sqrt{3}\right)^{2}}{4\left(\sqrt{7}+\sqrt{3}\right)}$$

The square of \sqrt{3} is 3.

$$\frac{7-3}{4\left(\sqrt{7}+\sqrt{3}\right)}$$

Subtract 3 from 7 to get 4.

$$\frac{4}{4\left(\sqrt{7}+\sqrt{3}\right)}$$

Use the distributive property to multiply 4 by \sqrt{7}+\sqrt{3}.

$$\frac{4}{4\sqrt{7}+4\sqrt{3}}$$

Rationalize the denominator of \frac{4}{4\sqrt{7}+4\sqrt{3}} by multiplying numerator and denominator by 4\sqrt{7}-4\sqrt{3}.

$$\frac{4\left(4\sqrt{7}-4\sqrt{3}\right)}{\left(4\sqrt{7}+4\sqrt{3}\right)\left(4\sqrt{7}-4\sqrt{3}\right)}$$

Consider \left(4\sqrt{7}+4\sqrt{3}\right)\left(4\sqrt{7}-4\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{4\left(4\sqrt{7}-4\sqrt{3}\right)}{\left(4\sqrt{7}\right)^{2}-\left(4\sqrt{3}\right)^{2}}$$

Expand \left(4\sqrt{7}\right)^{2}.

$$\frac{4\left(4\sqrt{7}-4\sqrt{3}\right)}{4^{2}\left(\sqrt{7}\right)^{2}-\left(4\sqrt{3}\right)^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{4\left(4\sqrt{7}-4\sqrt{3}\right)}{16\left(\sqrt{7}\right)^{2}-\left(4\sqrt{3}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$\frac{4\left(4\sqrt{7}-4\sqrt{3}\right)}{16\times 7-\left(4\sqrt{3}\right)^{2}}$$

Multiply 16 and 7 to get 112.

$$\frac{4\left(4\sqrt{7}-4\sqrt{3}\right)}{112-\left(4\sqrt{3}\right)^{2}}$$

Expand \left(4\sqrt{3}\right)^{2}.

$$\frac{4\left(4\sqrt{7}-4\sqrt{3}\right)}{112-4^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{4\left(4\sqrt{7}-4\sqrt{3}\right)}{112-16\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{4\left(4\sqrt{7}-4\sqrt{3}\right)}{112-16\times 3}$$

Multiply 16 and 3 to get 48.

$$\frac{4\left(4\sqrt{7}-4\sqrt{3}\right)}{112-48}$$

Subtract 48 from 112 to get 64.

$$\frac{4\left(4\sqrt{7}-4\sqrt{3}\right)}{64}$$

Divide 4\left(4\sqrt{7}-4\sqrt{3}\right) by 64 to get \frac{1}{16}\left(4\sqrt{7}-4\sqrt{3}\right).

$$\frac{1}{16}\left(4\sqrt{7}-4\sqrt{3}\right)$$

Use the distributive property to multiply \frac{1}{16} by 4\sqrt{7}-4\sqrt{3}.

$$\frac{1}{4}\sqrt{7}-\frac{1}{4}\sqrt{3}$$

Factor

$$\frac{\sqrt{7} - \sqrt{3}}{4} = 0.22842512587392838$$