Solve 5/7+3sqrt{2}

$$\frac { 5 } { 7 + 3 \sqrt { 2 } }$$

Evaluate

$$\frac{35-15\sqrt{2}}{31}\approx 0.444735373$$

Solution Steps

Rationalize the denominator of \frac{5}{7+3\sqrt{2}} by multiplying numerator and denominator by 7-3\sqrt{2}.

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

Consider \left(7+3\sqrt{2}\right)\left(7-3\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}.

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

Calculate 7 to the power of 2 and get 49.

$$\frac{5\left(7-3\sqrt{2}\right)}{49-\left(3\sqrt{2}\right)^{2}}$$

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

$$\frac{5\left(7-3\sqrt{2}\right)}{49-3^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{5\left(7-3\sqrt{2}\right)}{49-9\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{5\left(7-3\sqrt{2}\right)}{49-9\times 2}$$

Multiply 9 and 2 to get 18.

$$\frac{5\left(7-3\sqrt{2}\right)}{49-18}$$

Subtract 18 from 49 to get 31.

$$\frac{5\left(7-3\sqrt{2}\right)}{31}$$

Use the distributive property to multiply 5 by 7-3\sqrt{2}.

$$\frac{35-15\sqrt{2}}{31}$$