Solve |18|/sqrt{18sqrt{56}}

$$\frac { | 18 | } { \sqrt { 18 } \sqrt { 56 } }$$

Evaluate

$$\frac{3\sqrt{7}}{14}\approx 0.56694671$$

Solution Steps

The absolute value of a real number a is a when a\geq 0, or -a when a<0. The absolute value of 18 is 18.

$$\frac{18}{\sqrt{18}\sqrt{56}}$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$\frac{18}{3\sqrt{2}\sqrt{56}}$$

Factor 56=2^{2}\times 14. Rewrite the square root of the product \sqrt{2^{2}\times 14} as the product of square roots \sqrt{2^{2}}\sqrt{14}. Take the square root of 2^{2}.

$$\frac{18}{3\sqrt{2}\times 2\sqrt{14}}$$

Multiply 3 and 2 to get 6.

$$\frac{18}{6\sqrt{2}\sqrt{14}}$$

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}.

$$\frac{18}{6\sqrt{2}\sqrt{2}\sqrt{7}}$$

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

$$\frac{18}{6\times 2\sqrt{7}}$$

Rationalize the denominator of \frac{18}{6\times 2\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$\frac{18\sqrt{7}}{6\times 2\left(\sqrt{7}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$\frac{18\sqrt{7}}{6\times 2\times 7}$$

Cancel out 6 in both numerator and denominator.

$$\frac{3\sqrt{7}}{2\times 7}$$

Multiply 2 and 7 to get 14.

$$\frac{3\sqrt{7}}{14}$$