Solve 4/2+sqrt{10}

$$\frac { 4 } { 2 + \sqrt { 10 } }$$

Evaluate

$$\frac{2\left(\sqrt{10}-2\right)}{3}\approx 0.774851773$$

Solution Steps

Rationalize the denominator of \frac{4}{2+\sqrt{10}} by multiplying numerator and denominator by 2-\sqrt{10}.

$$\frac{4\left(2-\sqrt{10}\right)}{\left(2+\sqrt{10}\right)\left(2-\sqrt{10}\right)}$$

Consider \left(2+\sqrt{10}\right)\left(2-\sqrt{10}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{4\left(2-\sqrt{10}\right)}{2^{2}-\left(\sqrt{10}\right)^{2}}$$

Square 2. Square \sqrt{10}.

$$\frac{4\left(2-\sqrt{10}\right)}{4-10}$$

Subtract 10 from 4 to get -6.

$$\frac{4\left(2-\sqrt{10}\right)}{-6}$$

Divide 4\left(2-\sqrt{10}\right) by -6 to get -\frac{2}{3}\left(2-\sqrt{10}\right).

$$-\frac{2}{3}\left(2-\sqrt{10}\right)$$

Use the distributive property to multiply -\frac{2}{3} by 2-\sqrt{10}.

$$-\frac{2}{3}\times 2-\frac{2}{3}\left(-1\right)\sqrt{10}$$

Express -\frac{2}{3}\times 2 as a single fraction.

$$\frac{-2\times 2}{3}-\frac{2}{3}\left(-1\right)\sqrt{10}$$

Multiply -2 and 2 to get -4.

$$\frac{-4}{3}-\frac{2}{3}\left(-1\right)\sqrt{10}$$

Fraction \frac{-4}{3} can be rewritten as -\frac{4}{3} by extracting the negative sign.

$$-\frac{4}{3}-\frac{2}{3}\left(-1\right)\sqrt{10}$$

Multiply -\frac{2}{3} and -1 to get \frac{2}{3}.

$$-\frac{4}{3}+\frac{2}{3}\sqrt{10}$$