Solve 3sqrt{1-2/3+{left(1/2right)}^3}

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

Evaluate

$$\frac{\sqrt{66}}{4}\approx 2.031009601$$

Solution Steps

Convert 1 to fraction \frac{3}{3}.

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

Since \frac{3}{3} and \frac{2}{3} have the same denominator, subtract them by subtracting their numerators.

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

Subtract 2 from 3 to get 1.

$$3\sqrt{\frac{1}{3}+\left(\frac{1}{2}\right)^{3}}$$

Calculate \frac{1}{2} to the power of 3 and get \frac{1}{8}.

$$3\sqrt{\frac{1}{3}+\frac{1}{8}}$$

Least common multiple of 3 and 8 is 24. Convert \frac{1}{3} and \frac{1}{8} to fractions with denominator 24.

$$3\sqrt{\frac{8}{24}+\frac{3}{24}}$$

Since \frac{8}{24} and \frac{3}{24} have the same denominator, add them by adding their numerators.

$$3\sqrt{\frac{8+3}{24}}$$

Add 8 and 3 to get 11.

$$3\sqrt{\frac{11}{24}}$$

Rewrite the square root of the division \sqrt{\frac{11}{24}} as the division of square roots \frac{\sqrt{11}}{\sqrt{24}}.

$$3\times \frac{\sqrt{11}}{\sqrt{24}}$$

Factor 24=2^{2}\times 6. Rewrite the square root of the product \sqrt{2^{2}\times 6} as the product of square roots \sqrt{2^{2}}\sqrt{6}. Take the square root of 2^{2}.

$$3\times \frac{\sqrt{11}}{2\sqrt{6}}$$

Rationalize the denominator of \frac{\sqrt{11}}{2\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$3\times \frac{\sqrt{11}\sqrt{6}}{2\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$3\times \frac{\sqrt{11}\sqrt{6}}{2\times 6}$$

To multiply \sqrt{11} and \sqrt{6}, multiply the numbers under the square root.

$$3\times \frac{\sqrt{66}}{2\times 6}$$

Multiply 2 and 6 to get 12.

$$3\times \frac{\sqrt{66}}{12}$$

Cancel out 12, the greatest common factor in 3 and 12.

$$\frac{\sqrt{66}}{4}$$