Solve 9sqrt{45}divsqrt{11/2}

$$9 \sqrt { 45 } \div \sqrt { 1 \frac { 1 } { 2 } }$$

Evaluate

$$9\sqrt{30}\approx 49.295030175$$

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}.

$$\frac{9\times 3\sqrt{5}}{\sqrt{\frac{1\times 2+1}{2}}}$$

Multiply 9 and 3 to get 27.

$$\frac{27\sqrt{5}}{\sqrt{\frac{1\times 2+1}{2}}}$$

Multiply 1 and 2 to get 2.

$$\frac{27\sqrt{5}}{\sqrt{\frac{2+1}{2}}}$$

Add 2 and 1 to get 3.

$$\frac{27\sqrt{5}}{\sqrt{\frac{3}{2}}}$$

Rewrite the square root of the division \sqrt{\frac{3}{2}} as the division of square roots \frac{\sqrt{3}}{\sqrt{2}}.

$$\frac{27\sqrt{5}}{\frac{\sqrt{3}}{\sqrt{2}}}$$

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

$$\frac{27\sqrt{5}}{\frac{\sqrt{3}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}}$$

The square of \sqrt{2} is 2.

$$\frac{27\sqrt{5}}{\frac{\sqrt{3}\sqrt{2}}{2}}$$

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

$$\frac{27\sqrt{5}}{\frac{\sqrt{6}}{2}}$$

Divide 27\sqrt{5} by \frac{\sqrt{6}}{2} by multiplying 27\sqrt{5} by the reciprocal of \frac{\sqrt{6}}{2}.

$$\frac{27\sqrt{5}\times 2}{\sqrt{6}}$$

Rationalize the denominator of \frac{27\sqrt{5}\times 2}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{27\sqrt{5}\times 2\sqrt{6}}{\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{27\sqrt{5}\times 2\sqrt{6}}{6}$$

Multiply 27 and 2 to get 54.

$$\frac{54\sqrt{5}\sqrt{6}}{6}$$

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

$$\frac{54\sqrt{30}}{6}$$

Divide 54\sqrt{30} by 6 to get 9\sqrt{30}.

$$9\sqrt{30}$$