Solve frac{sqrt{80}*sqrt{54}}{sqrt{60}}

$$\frac { \sqrt { 80 } \times \sqrt { 54 } } { \sqrt { 60 } }$$

Evaluate

$$6\sqrt{2}\approx 8.485281374$$

Solution Steps

Factor 80=4^{2}\times 5. Rewrite the square root of the product \sqrt{4^{2}\times 5} as the product of square roots \sqrt{4^{2}}\sqrt{5}. Take the square root of 4^{2}.

$$\frac{4\sqrt{5}\sqrt{54}}{\sqrt{60}}$$

Factor 54=3^{2}\times 6. Rewrite the square root of the product \sqrt{3^{2}\times 6} as the product of square roots \sqrt{3^{2}}\sqrt{6}. Take the square root of 3^{2}.

$$\frac{4\sqrt{5}\times 3\sqrt{6}}{\sqrt{60}}$$

Multiply 4 and 3 to get 12.

$$\frac{12\sqrt{5}\sqrt{6}}{\sqrt{60}}$$

To multiply \sqrt{5} and \sqrt{6}, multiply the numbers under the square root.

$$\frac{12\sqrt{30}}{\sqrt{60}}$$

Factor 60=2^{2}\times 15. Rewrite the square root of the product \sqrt{2^{2}\times 15} as the product of square roots \sqrt{2^{2}}\sqrt{15}. Take the square root of 2^{2}.

$$\frac{12\sqrt{30}}{2\sqrt{15}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{6\sqrt{30}}{\sqrt{15}}$$

Rationalize the denominator of \frac{6\sqrt{30}}{\sqrt{15}} by multiplying numerator and denominator by \sqrt{15}.

$$\frac{6\sqrt{30}\sqrt{15}}{\left(\sqrt{15}\right)^{2}}$$

The square of \sqrt{15} is 15.

$$\frac{6\sqrt{30}\sqrt{15}}{15}$$

Factor 30=15\times 2. Rewrite the square root of the product \sqrt{15\times 2} as the product of square roots \sqrt{15}\sqrt{2}.

$$\frac{6\sqrt{15}\sqrt{2}\sqrt{15}}{15}$$

Multiply \sqrt{15} and \sqrt{15} to get 15.

$$\frac{6\times 15\sqrt{2}}{15}$$

Cancel out 15 and 15.

$$6\sqrt{2}$$