Solve sqrt{45}divsqrt{32}*sqrt{8}

$$\sqrt { 45 } \div \sqrt { 32 } \cdot \sqrt { 8 }$$

Evaluate

$$\frac{3\sqrt{5}}{2}\approx 3.354101966$$

Solution Steps

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

$$\frac{3\sqrt{5}}{\sqrt{32}}\sqrt{8}$$

Factor 32=4^{2}\times 2. Rewrite the square root of the product \sqrt{4^{2}\times 2} as the product of square roots \sqrt{4^{2}}\sqrt{2}. Take the square root of 4^{2}.

$$\frac{3\sqrt{5}}{4\sqrt{2}}\sqrt{8}$$

Rationalize the denominator of \frac{3\sqrt{5}}{4\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{3\sqrt{5}\sqrt{2}}{4\left(\sqrt{2}\right)^{2}}\sqrt{8}$$

The square of \sqrt{2} is 2.

$$\frac{3\sqrt{5}\sqrt{2}}{4\times 2}\sqrt{8}$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{3\sqrt{10}}{4\times 2}\sqrt{8}$$

Multiply 4 and 2 to get 8.

$$\frac{3\sqrt{10}}{8}\sqrt{8}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{3\sqrt{10}}{8}\times 2\sqrt{2}$$

Cancel out 8, the greatest common factor in 2 and 8.

$$\frac{3\sqrt{10}}{4}\sqrt{2}$$

Express \frac{3\sqrt{10}}{4}\sqrt{2} as a single fraction.

$$\frac{3\sqrt{10}\sqrt{2}}{4}$$

Factor 10=2\times 5. Rewrite the square root of the product \sqrt{2\times 5} as the product of square roots \sqrt{2}\sqrt{5}.

$$\frac{3\sqrt{2}\sqrt{5}\sqrt{2}}{4}$$

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

$$\frac{3\times 2\sqrt{5}}{4}$$

Multiply 3 and 2 to get 6.

$$\frac{6\sqrt{5}}{4}$$

Divide 6\sqrt{5} by 4 to get \frac{3}{2}\sqrt{5}.

$$\frac{3}{2}\sqrt{5}$$