Solve frac{sqrt{5*10^3}}{sqrt{150}}

$$\frac{ \sqrt{ 5 \times { 10 }^{ 3 } } }{ \sqrt{ 150 } }$$

Evaluate

$$\frac{10\sqrt{3}}{3}\approx 5.773502692$$

Solution Steps

Calculate 10 to the power of 3 and get 1000.

$$\frac{\sqrt{5\times 1000}}{\sqrt{150}}$$

Multiply 5 and 1000 to get 5000.

$$\frac{\sqrt{5000}}{\sqrt{150}}$$

Factor 5000=50^{2}\times 2. Rewrite the square root of the product \sqrt{50^{2}\times 2} as the product of square roots \sqrt{50^{2}}\sqrt{2}. Take the square root of 50^{2}.

$$\frac{50\sqrt{2}}{\sqrt{150}}$$

Factor 150=5^{2}\times 6. Rewrite the square root of the product \sqrt{5^{2}\times 6} as the product of square roots \sqrt{5^{2}}\sqrt{6}. Take the square root of 5^{2}.

$$\frac{50\sqrt{2}}{5\sqrt{6}}$$

Cancel out 5 in both numerator and denominator.

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

Rationalize the denominator of \frac{10\sqrt{2}}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

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

The square of \sqrt{6} is 6.

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

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

$$\frac{10\sqrt{2}\sqrt{2}\sqrt{3}}{6}$$

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

$$\frac{10\times 2\sqrt{3}}{6}$$

Multiply 10 and 2 to get 20.

$$\frac{20\sqrt{3}}{6}$$

Divide 20\sqrt{3} by 6 to get \frac{10}{3}\sqrt{3}.

$$\frac{10}{3}\sqrt{3}$$