Solve 4sqrt{6/16}+4sqrt{20}-9sqrt{3/27}

$$4 \sqrt { \frac { 6 } { 16 } } + 4 \sqrt { 20 } - 9 \sqrt { \frac { 3 } { 27 } }$$

Evaluate

$$\sqrt{6}+8\sqrt{5}-3\approx 17.338033563$$

Solution Steps

Reduce the fraction \frac{6}{16} to lowest terms by extracting and canceling out 2.

$$4\sqrt{\frac{3}{8}}+4\sqrt{20}-9\sqrt{\frac{3}{27}}$$

Rewrite the square root of the division \sqrt{\frac{3}{8}} as the division of square roots \frac{\sqrt{3}}{\sqrt{8}}.

$$4\times \frac{\sqrt{3}}{\sqrt{8}}+4\sqrt{20}-9\sqrt{\frac{3}{27}}$$

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}.

$$4\times \frac{\sqrt{3}}{2\sqrt{2}}+4\sqrt{20}-9\sqrt{\frac{3}{27}}$$

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

$$4\times \frac{\sqrt{3}\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}+4\sqrt{20}-9\sqrt{\frac{3}{27}}$$

The square of \sqrt{2} is 2.

$$4\times \frac{\sqrt{3}\sqrt{2}}{2\times 2}+4\sqrt{20}-9\sqrt{\frac{3}{27}}$$

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

$$4\times \frac{\sqrt{6}}{2\times 2}+4\sqrt{20}-9\sqrt{\frac{3}{27}}$$

Multiply 2 and 2 to get 4.

$$4\times \frac{\sqrt{6}}{4}+4\sqrt{20}-9\sqrt{\frac{3}{27}}$$

Cancel out 4 and 4.

$$\sqrt{6}+4\sqrt{20}-9\sqrt{\frac{3}{27}}$$

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$\sqrt{6}+4\times 2\sqrt{5}-9\sqrt{\frac{3}{27}}$$

Multiply 4 and 2 to get 8.

$$\sqrt{6}+8\sqrt{5}-9\sqrt{\frac{3}{27}}$$

Reduce the fraction \frac{3}{27} to lowest terms by extracting and canceling out 3.

$$\sqrt{6}+8\sqrt{5}-9\sqrt{\frac{1}{9}}$$

Rewrite the square root of the division \frac{1}{9} as the division of square roots \frac{\sqrt{1}}{\sqrt{9}}. Take the square root of both numerator and denominator.

$$\sqrt{6}+8\sqrt{5}-9\times \frac{1}{3}$$

Multiply 9 and \frac{1}{3} to get \frac{9}{3}.

$$\sqrt{6}+8\sqrt{5}-\frac{9}{3}$$

Divide 9 by 3 to get 3.

$$\sqrt{6}+8\sqrt{5}-3$$