Solve frac{sqrt{3}}{7-4sqrt{2}}

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

Evaluate

$$\frac{4\sqrt{6}+7\sqrt{3}}{17}\approx 1.289547919$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{3}}{7-4\sqrt{2}} by multiplying numerator and denominator by 7+4\sqrt{2}.

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

Consider \left(7-4\sqrt{2}\right)\left(7+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}.

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

Calculate 7 to the power of 2 and get 49.

$$\frac{\sqrt{3}\left(7+4\sqrt{2}\right)}{49-\left(-4\sqrt{2}\right)^{2}}$$

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

$$\frac{\sqrt{3}\left(7+4\sqrt{2}\right)}{49-\left(-4\right)^{2}\left(\sqrt{2}\right)^{2}}$$

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

$$\frac{\sqrt{3}\left(7+4\sqrt{2}\right)}{49-16\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{3}\left(7+4\sqrt{2}\right)}{49-16\times 2}$$

Multiply 16 and 2 to get 32.

$$\frac{\sqrt{3}\left(7+4\sqrt{2}\right)}{49-32}$$

Subtract 32 from 49 to get 17.

$$\frac{\sqrt{3}\left(7+4\sqrt{2}\right)}{17}$$

Use the distributive property to multiply \sqrt{3} by 7+4\sqrt{2}.

$$\frac{7\sqrt{3}+4\sqrt{3}\sqrt{2}}{17}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{7\sqrt{3}+4\sqrt{6}}{17}$$