Solve [sqrt{25}+sqrt[3]{(-8)^2}-(121/sqrt{3}+2)]sqrt{7-1/4}

$$[ \sqrt { 25 } + \sqrt[ 3 ] { ( - 8 ) ^ { 2 } } - ( \frac { 121 } { \sqrt { 3 } } + 2 ) ] \sqrt { 7 - \frac { 1 } { 4 } }$$

Evaluate

$$\frac{21\sqrt{3}-363}{2}\approx -163.313466521$$

Short Solution Steps

Calculate the square root of 25 and get 5.

$$\left(5+\sqrt[3]{\left(-8\right)^{2}}-\left(\frac{121}{\sqrt{3}}+2\right)\right)\sqrt{7-\frac{1}{4}}$$

Calculate -8 to the power of 2 and get 64.

$$\left(5+\sqrt[3]{64}-\left(\frac{121}{\sqrt{3}}+2\right)\right)\sqrt{7-\frac{1}{4}}$$

Calculate \sqrt[3]{64} and get 4.

$$\left(5+4-\left(\frac{121}{\sqrt{3}}+2\right)\right)\sqrt{7-\frac{1}{4}}$$

Add 5 and 4 to get 9.

$$\left(9-\left(\frac{121}{\sqrt{3}}+2\right)\right)\sqrt{7-\frac{1}{4}}$$

Rationalize the denominator of \frac{121}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\left(9-\left(\frac{121\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+2\right)\right)\sqrt{7-\frac{1}{4}}$$

The square of \sqrt{3} is 3.

$$\left(9-\left(\frac{121\sqrt{3}}{3}+2\right)\right)\sqrt{7-\frac{1}{4}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{3}{3}.

$$\left(9-\left(\frac{121\sqrt{3}}{3}+\frac{2\times 3}{3}\right)\right)\sqrt{7-\frac{1}{4}}$$

Since \frac{121\sqrt{3}}{3} and \frac{2\times 3}{3} have the same denominator, add them by adding their numerators.

$$\left(9-\frac{121\sqrt{3}+2\times 3}{3}\right)\sqrt{7-\frac{1}{4}}$$

Do the multiplications in 121\sqrt{3}+2\times 3.

$$\left(9-\frac{121\sqrt{3}+6}{3}\right)\sqrt{7-\frac{1}{4}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 9 times \frac{3}{3}.

$$\left(\frac{9\times 3}{3}-\frac{121\sqrt{3}+6}{3}\right)\sqrt{7-\frac{1}{4}}$$

Since \frac{9\times 3}{3} and \frac{121\sqrt{3}+6}{3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{9\times 3-\left(121\sqrt{3}+6\right)}{3}\sqrt{7-\frac{1}{4}}$$

Do the multiplications in 9\times 3-\left(121\sqrt{3}+6\right).

$$\frac{27-121\sqrt{3}-6}{3}\sqrt{7-\frac{1}{4}}$$

Do the calculations in 27-121\sqrt{3}-6.

$$\frac{21-121\sqrt{3}}{3}\sqrt{7-\frac{1}{4}}$$

Subtract \frac{1}{4} from 7 to get \frac{27}{4}.

$$\frac{21-121\sqrt{3}}{3}\sqrt{\frac{27}{4}}$$

Rewrite the square root of the division \sqrt{\frac{27}{4}} as the division of square roots \frac{\sqrt{27}}{\sqrt{4}}.

$$\frac{21-121\sqrt{3}}{3}\times \frac{\sqrt{27}}{\sqrt{4}}$$

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}.

$$\frac{21-121\sqrt{3}}{3}\times \frac{3\sqrt{3}}{\sqrt{4}}$$

Calculate the square root of 4 and get 2.

$$\frac{21-121\sqrt{3}}{3}\times \frac{3\sqrt{3}}{2}$$

Multiply \frac{21-121\sqrt{3}}{3} times \frac{3\sqrt{3}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(21-121\sqrt{3}\right)\times 3\sqrt{3}}{3\times 2}$$

Cancel out 3 in both numerator and denominator.

$$\frac{\sqrt{3}\left(-121\sqrt{3}+21\right)}{2}$$

Use the distributive property to multiply \sqrt{3} by -121\sqrt{3}+21.

$$\frac{-121\left(\sqrt{3}\right)^{2}+21\sqrt{3}}{2}$$

The square of \sqrt{3} is 3.

$$\frac{-121\times 3+21\sqrt{3}}{2}$$

Multiply -121 and 3 to get -363.

$$\frac{-363+21\sqrt{3}}{2}$$