Solve frac{sqrt{108}*sqrt{600}}{sqrt{675}}

$$\frac { \sqrt { 108 } \times \sqrt { 600 } } { \sqrt { 675 } }$$

Evaluate

$$4\sqrt{6}\approx 9.797958971$$

Solution Steps

Factor 108=6^{2}\times 3. Rewrite the square root of the product \sqrt{6^{2}\times 3} as the product of square roots \sqrt{6^{2}}\sqrt{3}. Take the square root of 6^{2}.

$$\frac{6\sqrt{3}\sqrt{600}}{\sqrt{675}}$$

Factor 600=10^{2}\times 6. Rewrite the square root of the product \sqrt{10^{2}\times 6} as the product of square roots \sqrt{10^{2}}\sqrt{6}. Take the square root of 10^{2}.

$$\frac{6\sqrt{3}\times 10\sqrt{6}}{\sqrt{675}}$$

Multiply 6 and 10 to get 60.

$$\frac{60\sqrt{3}\sqrt{6}}{\sqrt{675}}$$

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

$$\frac{60\sqrt{3}\sqrt{3}\sqrt{2}}{\sqrt{675}}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{60\times 3\sqrt{2}}{\sqrt{675}}$$

Factor 675=15^{2}\times 3. Rewrite the square root of the product \sqrt{15^{2}\times 3} as the product of square roots \sqrt{15^{2}}\sqrt{3}. Take the square root of 15^{2}.

$$\frac{60\times 3\sqrt{2}}{15\sqrt{3}}$$

Cancel out 15 in both numerator and denominator.

$$\frac{3\times 4\sqrt{2}}{\sqrt{3}}$$

Rationalize the denominator of \frac{3\times 4\sqrt{2}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{3\times 4\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{3\times 4\sqrt{2}\sqrt{3}}{3}$$

Multiply 3 and 4 to get 12.

$$\frac{12\sqrt{2}\sqrt{3}}{3}$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{12\sqrt{6}}{3}$$

Divide 12\sqrt{6} by 3 to get 4\sqrt{6}.

$$4\sqrt{6}$$