Solve frac{6sqrt{14}-4sqrt{2}}{2sqrt{2}}

$$\frac { 6 \sqrt { 14 } - 4 \sqrt { 2 } } { 2 \sqrt { 2 } }$$

Evaluate

$$3\sqrt{7}-2\approx 5.937253933$$

Solution Steps

Rationalize the denominator of \frac{6\sqrt{14}-4\sqrt{2}}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\left(6\sqrt{14}-4\sqrt{2}\right)\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(6\sqrt{14}-4\sqrt{2}\right)\sqrt{2}}{2\times 2}$$

Multiply 2 and 2 to get 4.

$$\frac{\left(6\sqrt{14}-4\sqrt{2}\right)\sqrt{2}}{4}$$

Use the distributive property to multiply 6\sqrt{14}-4\sqrt{2} by \sqrt{2}.

$$\frac{6\sqrt{14}\sqrt{2}-4\left(\sqrt{2}\right)^{2}}{4}$$

Factor 14=2\times 7. Rewrite the square root of the product \sqrt{2\times 7} as the product of square roots \sqrt{2}\sqrt{7}.

$$\frac{6\sqrt{2}\sqrt{7}\sqrt{2}-4\left(\sqrt{2}\right)^{2}}{4}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{6\times 2\sqrt{7}-4\left(\sqrt{2}\right)^{2}}{4}$$

Multiply 6 and 2 to get 12.

$$\frac{12\sqrt{7}-4\left(\sqrt{2}\right)^{2}}{4}$$

The square of \sqrt{2} is 2.

$$\frac{12\sqrt{7}-4\times 2}{4}$$

Multiply -4 and 2 to get -8.

$$\frac{12\sqrt{7}-8}{4}$$

Divide each term of 12\sqrt{7}-8 by 4 to get 3\sqrt{7}-2.

$$3\sqrt{7}-2$$

Factor

$$3 \sqrt{7} - 2 = 5.937253933$$