Solve 2/5sqrt{3+3sqrt{15}}

$$\frac { 2 } { 5 \sqrt { 3 } + 3 \sqrt { 15 } }$$

Evaluate

$$\frac{\sqrt{3}\left(3\sqrt{5}-5\right)}{30}\approx 0.0986232$$

Solution Steps

Rationalize the denominator of \frac{2}{5\sqrt{3}+3\sqrt{15}} by multiplying numerator and denominator by 5\sqrt{3}-3\sqrt{15}.

$$\frac{2\left(5\sqrt{3}-3\sqrt{15}\right)}{\left(5\sqrt{3}+3\sqrt{15}\right)\left(5\sqrt{3}-3\sqrt{15}\right)}$$

Consider \left(5\sqrt{3}+3\sqrt{15}\right)\left(5\sqrt{3}-3\sqrt{15}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{2\left(5\sqrt{3}-3\sqrt{15}\right)}{\left(5\sqrt{3}\right)^{2}-\left(3\sqrt{15}\right)^{2}}$$

Expand \left(5\sqrt{3}\right)^{2}.

$$\frac{2\left(5\sqrt{3}-3\sqrt{15}\right)}{5^{2}\left(\sqrt{3}\right)^{2}-\left(3\sqrt{15}\right)^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{2\left(5\sqrt{3}-3\sqrt{15}\right)}{25\left(\sqrt{3}\right)^{2}-\left(3\sqrt{15}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{2\left(5\sqrt{3}-3\sqrt{15}\right)}{25\times 3-\left(3\sqrt{15}\right)^{2}}$$

Multiply 25 and 3 to get 75.

$$\frac{2\left(5\sqrt{3}-3\sqrt{15}\right)}{75-\left(3\sqrt{15}\right)^{2}}$$

Expand \left(3\sqrt{15}\right)^{2}.

$$\frac{2\left(5\sqrt{3}-3\sqrt{15}\right)}{75-3^{2}\left(\sqrt{15}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{2\left(5\sqrt{3}-3\sqrt{15}\right)}{75-9\left(\sqrt{15}\right)^{2}}$$

The square of \sqrt{15} is 15.

$$\frac{2\left(5\sqrt{3}-3\sqrt{15}\right)}{75-9\times 15}$$

Multiply 9 and 15 to get 135.

$$\frac{2\left(5\sqrt{3}-3\sqrt{15}\right)}{75-135}$$

Subtract 135 from 75 to get -60.

$$\frac{2\left(5\sqrt{3}-3\sqrt{15}\right)}{-60}$$

Divide 2\left(5\sqrt{3}-3\sqrt{15}\right) by -60 to get -\frac{1}{30}\left(5\sqrt{3}-3\sqrt{15}\right).

$$-\frac{1}{30}\left(5\sqrt{3}-3\sqrt{15}\right)$$

Use the distributive property to multiply -\frac{1}{30} by 5\sqrt{3}-3\sqrt{15}.

$$-\frac{1}{30}\times 5\sqrt{3}-\frac{1}{30}\left(-3\right)\sqrt{15}$$

Express -\frac{1}{30}\times 5 as a single fraction.

$$\frac{-5}{30}\sqrt{3}-\frac{1}{30}\left(-3\right)\sqrt{15}$$

Reduce the fraction \frac{-5}{30} to lowest terms by extracting and canceling out 5.

$$-\frac{1}{6}\sqrt{3}-\frac{1}{30}\left(-3\right)\sqrt{15}$$

Express -\frac{1}{30}\left(-3\right) as a single fraction.

$$-\frac{1}{6}\sqrt{3}+\frac{-\left(-3\right)}{30}\sqrt{15}$$

Multiply -1 and -3 to get 3.

$$-\frac{1}{6}\sqrt{3}+\frac{3}{30}\sqrt{15}$$

Reduce the fraction \frac{3}{30} to lowest terms by extracting and canceling out 3.

$$-\frac{1}{6}\sqrt{3}+\frac{1}{10}\sqrt{15}$$