Solve frac{(sqrt{3}+sqrt{7})(sqrt{3}-sqrt{7})}{2sqrt{5}-4sqrt{2}}

$$\frac { ( \sqrt { 3 } + \sqrt { 7 } ) ( \sqrt { 3 } - \sqrt { 7 } ) } { 2 \sqrt { 5 } - 4 \sqrt { 2 } }$$

Evaluate

$$\frac{2\left(\sqrt{5}+2\sqrt{2}\right)}{3}\approx 3.376330068$$

Solution Steps

Consider \left(\sqrt{3}+\sqrt{7}\right)\left(\sqrt{3}-\sqrt{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}.

$$\frac{\left(\sqrt{3}\right)^{2}-\left(\sqrt{7}\right)^{2}}{2\sqrt{5}-4\sqrt{2}}$$

The square of \sqrt{3} is 3.

$$\frac{3-\left(\sqrt{7}\right)^{2}}{2\sqrt{5}-4\sqrt{2}}$$

The square of \sqrt{7} is 7.

$$\frac{3-7}{2\sqrt{5}-4\sqrt{2}}$$

Subtract 7 from 3 to get -4.

$$\frac{-4}{2\sqrt{5}-4\sqrt{2}}$$

Rationalize the denominator of \frac{-4}{2\sqrt{5}-4\sqrt{2}} by multiplying numerator and denominator by 2\sqrt{5}+4\sqrt{2}.

$$\frac{-4\left(2\sqrt{5}+4\sqrt{2}\right)}{\left(2\sqrt{5}-4\sqrt{2}\right)\left(2\sqrt{5}+4\sqrt{2}\right)}$$

Consider \left(2\sqrt{5}-4\sqrt{2}\right)\left(2\sqrt{5}+4\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{-4\left(2\sqrt{5}+4\sqrt{2}\right)}{\left(2\sqrt{5}\right)^{2}-\left(-4\sqrt{2}\right)^{2}}$$

Expand \left(2\sqrt{5}\right)^{2}.

$$\frac{-4\left(2\sqrt{5}+4\sqrt{2}\right)}{2^{2}\left(\sqrt{5}\right)^{2}-\left(-4\sqrt{2}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{-4\left(2\sqrt{5}+4\sqrt{2}\right)}{4\left(\sqrt{5}\right)^{2}-\left(-4\sqrt{2}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{-4\left(2\sqrt{5}+4\sqrt{2}\right)}{4\times 5-\left(-4\sqrt{2}\right)^{2}}$$

Multiply 4 and 5 to get 20.

$$\frac{-4\left(2\sqrt{5}+4\sqrt{2}\right)}{20-\left(-4\sqrt{2}\right)^{2}}$$

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

$$\frac{-4\left(2\sqrt{5}+4\sqrt{2}\right)}{20-\left(-4\right)^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate -4 to the power of 2 and get 16.

$$\frac{-4\left(2\sqrt{5}+4\sqrt{2}\right)}{20-16\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{-4\left(2\sqrt{5}+4\sqrt{2}\right)}{20-16\times 2}$$

Multiply 16 and 2 to get 32.

$$\frac{-4\left(2\sqrt{5}+4\sqrt{2}\right)}{20-32}$$

Subtract 32 from 20 to get -12.

$$\frac{-4\left(2\sqrt{5}+4\sqrt{2}\right)}{-12}$$

Divide -4\left(2\sqrt{5}+4\sqrt{2}\right) by -12 to get \frac{1}{3}\left(2\sqrt{5}+4\sqrt{2}\right).

$$\frac{1}{3}\left(2\sqrt{5}+4\sqrt{2}\right)$$

Use the distributive property to multiply \frac{1}{3} by 2\sqrt{5}+4\sqrt{2}.

$$\frac{1}{3}\times 2\sqrt{5}+\frac{1}{3}\times 4\sqrt{2}$$

Multiply \frac{1}{3} and 2 to get \frac{2}{3}.

$$\frac{2}{3}\sqrt{5}+\frac{1}{3}\times 4\sqrt{2}$$

Multiply \frac{1}{3} and 4 to get \frac{4}{3}.

$$\frac{2}{3}\sqrt{5}+\frac{4}{3}\sqrt{2}$$