Solve (3sqrt{27}+1/5sqrt{75}-6sqrt{1/3})divsqrt{48}

$$( 3 \sqrt { 27 } + \frac { 1 } { 5 \sqrt { 75 } } - 6 \sqrt { \frac { 1 } { 3 } } ) \div \sqrt { 48 }$$

Evaluate

$$\frac{263}{150}\approx 1.753333333$$

Solution Steps

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$\frac{3\times 3\sqrt{3}+\frac{1}{5\sqrt{75}}-6\sqrt{\frac{1}{3}}}{\sqrt{48}}$$

Multiply 3 and 3 to get 9.

$$\frac{9\sqrt{3}+\frac{1}{5\sqrt{75}}-6\sqrt{\frac{1}{3}}}{\sqrt{48}}$$

Factor 75=5^{2}\times 3. Rewrite the square root of the product \sqrt{5^{2}\times 3} as the product of square roots \sqrt{5^{2}}\sqrt{3}. Take the square root of 5^{2}.

$$\frac{9\sqrt{3}+\frac{1}{5\times 5\sqrt{3}}-6\sqrt{\frac{1}{3}}}{\sqrt{48}}$$

Multiply 5 and 5 to get 25.

$$\frac{9\sqrt{3}+\frac{1}{25\sqrt{3}}-6\sqrt{\frac{1}{3}}}{\sqrt{48}}$$

Rationalize the denominator of \frac{1}{25\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{9\sqrt{3}+\frac{\sqrt{3}}{25\left(\sqrt{3}\right)^{2}}-6\sqrt{\frac{1}{3}}}{\sqrt{48}}$$

The square of \sqrt{3} is 3.

$$\frac{9\sqrt{3}+\frac{\sqrt{3}}{25\times 3}-6\sqrt{\frac{1}{3}}}{\sqrt{48}}$$

Multiply 25 and 3 to get 75.

$$\frac{9\sqrt{3}+\frac{\sqrt{3}}{75}-6\sqrt{\frac{1}{3}}}{\sqrt{48}}$$

Combine 9\sqrt{3} and \frac{\sqrt{3}}{75} to get \frac{676}{75}\sqrt{3}.

$$\frac{\frac{676}{75}\sqrt{3}-6\sqrt{\frac{1}{3}}}{\sqrt{48}}$$

Rewrite the square root of the division \sqrt{\frac{1}{3}} as the division of square roots \frac{\sqrt{1}}{\sqrt{3}}.

$$\frac{\frac{676}{75}\sqrt{3}-6\times \frac{\sqrt{1}}{\sqrt{3}}}{\sqrt{48}}$$

Calculate the square root of 1 and get 1.

$$\frac{\frac{676}{75}\sqrt{3}-6\times \frac{1}{\sqrt{3}}}{\sqrt{48}}$$

Rationalize the denominator of \frac{1}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\frac{676}{75}\sqrt{3}-6\times \frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}{\sqrt{48}}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{676}{75}\sqrt{3}-6\times \frac{\sqrt{3}}{3}}{\sqrt{48}}$$

Cancel out 3, the greatest common factor in 6 and 3.

$$\frac{\frac{676}{75}\sqrt{3}-2\sqrt{3}}{\sqrt{48}}$$

Combine \frac{676}{75}\sqrt{3} and -2\sqrt{3} to get \frac{526}{75}\sqrt{3}.

$$\frac{\frac{526}{75}\sqrt{3}}{\sqrt{48}}$$

Factor 48=4^{2}\times 3. Rewrite the square root of the product \sqrt{4^{2}\times 3} as the product of square roots \sqrt{4^{2}}\sqrt{3}. Take the square root of 4^{2}.

$$\frac{\frac{526}{75}\sqrt{3}}{4\sqrt{3}}$$

Cancel out \sqrt{3} in both numerator and denominator.

$$\frac{\frac{526}{75}}{4}$$

Express \frac{\frac{526}{75}}{4} as a single fraction.

$$\frac{526}{75\times 4}$$

Multiply 75 and 4 to get 300.

$$\frac{526}{300}$$

Reduce the fraction \frac{526}{300} to lowest terms by extracting and canceling out 2.

$$\frac{263}{150}$$

Factor

$$\frac{263}{2 \cdot 3 \cdot 5 ^ {2}} = 1\frac{113}{150} = 1.7533333333333334$$