Solve 3sqrt{147}-7/3sqrt{1/3}+7sqrt{1/3}=?

$$3 \sqrt { 147 } - \frac { 7 } { 3 } \sqrt { \frac { 1 } { 3 } } + 7 \sqrt { \frac { 1 } { 3 } } = ?$$

Evaluate

$$\frac{203\sqrt{3}}{9}\approx 39.067368215$$

Solution Steps

Factor 147=7^{2}\times 3. Rewrite the square root of the product \sqrt{7^{2}\times 3} as the product of square roots \sqrt{7^{2}}\sqrt{3}. Take the square root of 7^{2}.

$$3\times 7\sqrt{3}-\frac{7}{3}\sqrt{\frac{1}{3}}+7\sqrt{\frac{1}{3}}$$

Multiply 3 and 7 to get 21.

$$21\sqrt{3}-\frac{7}{3}\sqrt{\frac{1}{3}}+7\sqrt{\frac{1}{3}}$$

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

$$21\sqrt{3}-\frac{7}{3}\times \frac{\sqrt{1}}{\sqrt{3}}+7\sqrt{\frac{1}{3}}$$

Calculate the square root of 1 and get 1.

$$21\sqrt{3}-\frac{7}{3}\times \frac{1}{\sqrt{3}}+7\sqrt{\frac{1}{3}}$$

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

$$21\sqrt{3}-\frac{7}{3}\times \frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+7\sqrt{\frac{1}{3}}$$

The square of \sqrt{3} is 3.

$$21\sqrt{3}-\frac{7}{3}\times \frac{\sqrt{3}}{3}+7\sqrt{\frac{1}{3}}$$

Multiply -\frac{7}{3} times \frac{\sqrt{3}}{3} by multiplying numerator times numerator and denominator times denominator.

$$21\sqrt{3}+\frac{-7\sqrt{3}}{3\times 3}+7\sqrt{\frac{1}{3}}$$

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

$$21\sqrt{3}+\frac{-7\sqrt{3}}{3\times 3}+7\times \frac{\sqrt{1}}{\sqrt{3}}$$

Calculate the square root of 1 and get 1.

$$21\sqrt{3}+\frac{-7\sqrt{3}}{3\times 3}+7\times \frac{1}{\sqrt{3}}$$

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

$$21\sqrt{3}+\frac{-7\sqrt{3}}{3\times 3}+7\times \frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$21\sqrt{3}+\frac{-7\sqrt{3}}{3\times 3}+7\times \frac{\sqrt{3}}{3}$$

Express 7\times \frac{\sqrt{3}}{3} as a single fraction.

$$21\sqrt{3}+\frac{-7\sqrt{3}}{3\times 3}+\frac{7\sqrt{3}}{3}$$

Combine 21\sqrt{3} and \frac{7\sqrt{3}}{3} to get \frac{70}{3}\sqrt{3}.

$$\frac{70}{3}\sqrt{3}+\frac{-7\sqrt{3}}{3\times 3}$$

Multiply 3 and 3 to get 9.

$$\frac{70}{3}\sqrt{3}+\frac{-7\sqrt{3}}{9}$$

Combine \frac{70}{3}\sqrt{3} and \frac{-7\sqrt{3}}{9} to get \frac{203}{9}\sqrt{3}.

$$\frac{203}{9}\sqrt{3}$$