Solve frac{5sqrt{120}}{6sqrt{40}}

$$\frac{ 5 \sqrt{ 120 } }{ 6 \sqrt{ 40 } }$$

Evaluate

$$\frac{5\sqrt{3}}{6}\approx 1.443375673$$

Solution Steps

Factor 120=2^{2}\times 30. Rewrite the square root of the product \sqrt{2^{2}\times 30} as the product of square roots \sqrt{2^{2}}\sqrt{30}. Take the square root of 2^{2}.

$$\frac{5\times 2\sqrt{30}}{6\sqrt{40}}$$

Multiply 5 and 2 to get 10.

$$\frac{10\sqrt{30}}{6\sqrt{40}}$$

Factor 40=2^{2}\times 10. Rewrite the square root of the product \sqrt{2^{2}\times 10} as the product of square roots \sqrt{2^{2}}\sqrt{10}. Take the square root of 2^{2}.

$$\frac{10\sqrt{30}}{6\times 2\sqrt{10}}$$

Multiply 6 and 2 to get 12.

$$\frac{10\sqrt{30}}{12\sqrt{10}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{5\sqrt{30}}{6\sqrt{10}}$$

Rationalize the denominator of \frac{5\sqrt{30}}{6\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$\frac{5\sqrt{30}\sqrt{10}}{6\left(\sqrt{10}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$\frac{5\sqrt{30}\sqrt{10}}{6\times 10}$$

Factor 30=10\times 3. Rewrite the square root of the product \sqrt{10\times 3} as the product of square roots \sqrt{10}\sqrt{3}.

$$\frac{5\sqrt{10}\sqrt{3}\sqrt{10}}{6\times 10}$$

Multiply \sqrt{10} and \sqrt{10} to get 10.

$$\frac{5\times 10\sqrt{3}}{6\times 10}$$

Multiply 6 and 10 to get 60.

$$\frac{5\times 10\sqrt{3}}{60}$$

Multiply 5 and 10 to get 50.

$$\frac{50\sqrt{3}}{60}$$

Divide 50\sqrt{3} by 60 to get \frac{5}{6}\sqrt{3}.

$$\frac{5}{6}\sqrt{3}$$