Solve sqrt{21/4*{left(11/3right)}^2+1/sqrt[3]{3frac{3{8}}}}

$$\sqrt{ 2 \frac{ 1 }{ 4 } \times { \left(1 \frac{ 1 }{ 3 } \right) }^{ 2 } + \frac{ 1 }{ \sqrt[ 3 ]{ 3 \frac{ 3 }{ 8 } } } }$$

Evaluate

$$\frac{\sqrt{42}}{3}\approx 2.160246899$$

Solution Steps

Multiply 2 and 4 to get 8.

$$\sqrt{\frac{8+1}{4}\times \left(\frac{1\times 3+1}{3}\right)^{2}+\frac{1}{\sqrt[3]{\frac{3\times 8+3}{8}}}}$$

Add 8 and 1 to get 9.

$$\sqrt{\frac{9}{4}\times \left(\frac{1\times 3+1}{3}\right)^{2}+\frac{1}{\sqrt[3]{\frac{3\times 8+3}{8}}}}$$

Multiply 1 and 3 to get 3.

$$\sqrt{\frac{9}{4}\times \left(\frac{3+1}{3}\right)^{2}+\frac{1}{\sqrt[3]{\frac{3\times 8+3}{8}}}}$$

Add 3 and 1 to get 4.

$$\sqrt{\frac{9}{4}\times \left(\frac{4}{3}\right)^{2}+\frac{1}{\sqrt[3]{\frac{3\times 8+3}{8}}}}$$

Calculate \frac{4}{3} to the power of 2 and get \frac{16}{9}.

$$\sqrt{\frac{9}{4}\times \frac{16}{9}+\frac{1}{\sqrt[3]{\frac{3\times 8+3}{8}}}}$$

Multiply \frac{9}{4} and \frac{16}{9} to get 4.

$$\sqrt{4+\frac{1}{\sqrt[3]{\frac{3\times 8+3}{8}}}}$$

Multiply 3 and 8 to get 24.

$$\sqrt{4+\frac{1}{\sqrt[3]{\frac{24+3}{8}}}}$$

Add 24 and 3 to get 27.

$$\sqrt{4+\frac{1}{\sqrt[3]{\frac{27}{8}}}}$$

Calculate \sqrt[3]{\frac{27}{8}} and get \frac{3}{2}.

$$\sqrt{4+\frac{1}{\frac{3}{2}}}$$

Divide 1 by \frac{3}{2} by multiplying 1 by the reciprocal of \frac{3}{2}.

$$\sqrt{4+1\times \frac{2}{3}}$$

Multiply 1 and \frac{2}{3} to get \frac{2}{3}.

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

Add 4 and \frac{2}{3} to get \frac{14}{3}.

$$\sqrt{\frac{14}{3}}$$

Rewrite the square root of the division \sqrt{\frac{14}{3}} as the division of square roots \frac{\sqrt{14}}{\sqrt{3}}.

$$\frac{\sqrt{14}}{\sqrt{3}}$$

Rationalize the denominator of \frac{\sqrt{14}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\sqrt{14}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{14}\sqrt{3}}{3}$$

To multiply \sqrt{14} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\sqrt{42}}{3}$$