Solve frac{5sqrt{2}}{3sqrt{2}-4}

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

Evaluate

$$5\left(2\sqrt{2}+3\right)\approx 29.142135624$$

Solution Steps

Rationalize the denominator of \frac{5\sqrt{2}}{3\sqrt{2}-4} by multiplying numerator and denominator by 3\sqrt{2}+4.

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

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

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

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

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

Calculate 3 to the power of 2 and get 9.

$$\frac{5\sqrt{2}\left(3\sqrt{2}+4\right)}{9\left(\sqrt{2}\right)^{2}-4^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{5\sqrt{2}\left(3\sqrt{2}+4\right)}{9\times 2-4^{2}}$$

Multiply 9 and 2 to get 18.

$$\frac{5\sqrt{2}\left(3\sqrt{2}+4\right)}{18-4^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{5\sqrt{2}\left(3\sqrt{2}+4\right)}{18-16}$$

Subtract 16 from 18 to get 2.

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

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

$$\frac{15\left(\sqrt{2}\right)^{2}+20\sqrt{2}}{2}$$

The square of \sqrt{2} is 2.

$$\frac{15\times 2+20\sqrt{2}}{2}$$

Multiply 15 and 2 to get 30.

$$\frac{30+20\sqrt{2}}{2}$$