Solve 2/5-3sqrt{2}+2/5+3sqrt{2}

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

Evaluate

$$\frac{20}{7}\approx 2.857142857$$

Short Solution Steps

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

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

Consider \left(5-3\sqrt{2}\right)\left(5+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}.

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

Calculate 5 to the power of 2 and get 25.

$$\frac{2\left(5+3\sqrt{2}\right)}{25-\left(-3\sqrt{2}\right)^{2}}+\frac{2}{5+3\sqrt{2}}$$

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

$$\frac{2\left(5+3\sqrt{2}\right)}{25-\left(-3\right)^{2}\left(\sqrt{2}\right)^{2}}+\frac{2}{5+3\sqrt{2}}$$

Calculate -3 to the power of 2 and get 9.

$$\frac{2\left(5+3\sqrt{2}\right)}{25-9\left(\sqrt{2}\right)^{2}}+\frac{2}{5+3\sqrt{2}}$$

The square of \sqrt{2} is 2.

$$\frac{2\left(5+3\sqrt{2}\right)}{25-9\times 2}+\frac{2}{5+3\sqrt{2}}$$

Multiply 9 and 2 to get 18.

$$\frac{2\left(5+3\sqrt{2}\right)}{25-18}+\frac{2}{5+3\sqrt{2}}$$

Subtract 18 from 25 to get 7.

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

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

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

Consider \left(5+3\sqrt{2}\right)\left(5-3\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}.

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

Calculate 5 to the power of 2 and get 25.

$$\frac{2\left(5+3\sqrt{2}\right)}{7}+\frac{2\left(5-3\sqrt{2}\right)}{25-\left(3\sqrt{2}\right)^{2}}$$

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

$$\frac{2\left(5+3\sqrt{2}\right)}{7}+\frac{2\left(5-3\sqrt{2}\right)}{25-3^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{2\left(5+3\sqrt{2}\right)}{7}+\frac{2\left(5-3\sqrt{2}\right)}{25-9\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{2\left(5+3\sqrt{2}\right)}{7}+\frac{2\left(5-3\sqrt{2}\right)}{25-9\times 2}$$

Multiply 9 and 2 to get 18.

$$\frac{2\left(5+3\sqrt{2}\right)}{7}+\frac{2\left(5-3\sqrt{2}\right)}{25-18}$$

Subtract 18 from 25 to get 7.

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

Since \frac{2\left(5+3\sqrt{2}\right)}{7} and \frac{2\left(5-3\sqrt{2}\right)}{7} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 2\left(5+3\sqrt{2}\right)+2\left(5-3\sqrt{2}\right).

$$\frac{10+6\sqrt{2}+10-6\sqrt{2}}{7}$$

Do the calculations in 10+6\sqrt{2}+10-6\sqrt{2}.

$$\frac{20}{7}$$

Factor

$$\frac{2 ^ {2} \cdot 5}{7} = 2\frac{6}{7} = 2.857142857142857$$