Solve sqrt{(1/2)^2-(1/5)^2}*3/2div2/5-(frac{3}{4})^2*frac{4}{3}divfrac{3}{4}

$$\sqrt { ( \frac { 1 } { 2 } ) ^ { 2 } - ( \frac { 1 } { 5 } ) ^ { 2 } } \times \frac { 3 } { 2 } \div \frac { 2 } { 5 } - ( \frac { 3 } { 4 } ) ^ { 2 } \times \frac { 4 } { 3 } \div \frac { 3 } { 4 }$$

Evaluate

$$\frac{3\sqrt{21}}{8}-1\approx 0.718465886$$

Short Solution Steps

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\frac{\sqrt{\frac{1}{4}-\left(\frac{1}{5}\right)^{2}}\times \frac{3}{2}}{\frac{2}{5}}-\frac{\left(\frac{3}{4}\right)^{2}\times \frac{4}{3}}{\frac{3}{4}}$$

Calculate \frac{1}{5} to the power of 2 and get \frac{1}{25}.

$$\frac{\sqrt{\frac{1}{4}-\frac{1}{25}}\times \frac{3}{2}}{\frac{2}{5}}-\frac{\left(\frac{3}{4}\right)^{2}\times \frac{4}{3}}{\frac{3}{4}}$$

Least common multiple of 4 and 25 is 100. Convert \frac{1}{4} and \frac{1}{25} to fractions with denominator 100.

$$\frac{\sqrt{\frac{25}{100}-\frac{4}{100}}\times \frac{3}{2}}{\frac{2}{5}}-\frac{\left(\frac{3}{4}\right)^{2}\times \frac{4}{3}}{\frac{3}{4}}$$

Since \frac{25}{100} and \frac{4}{100} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\sqrt{\frac{25-4}{100}}\times \frac{3}{2}}{\frac{2}{5}}-\frac{\left(\frac{3}{4}\right)^{2}\times \frac{4}{3}}{\frac{3}{4}}$$

Subtract 4 from 25 to get 21.

$$\frac{\sqrt{\frac{21}{100}}\times \frac{3}{2}}{\frac{2}{5}}-\frac{\left(\frac{3}{4}\right)^{2}\times \frac{4}{3}}{\frac{3}{4}}$$

Rewrite the square root of the division \sqrt{\frac{21}{100}} as the division of square roots \frac{\sqrt{21}}{\sqrt{100}}.

$$\frac{\frac{\sqrt{21}}{\sqrt{100}}\times \frac{3}{2}}{\frac{2}{5}}-\frac{\left(\frac{3}{4}\right)^{2}\times \frac{4}{3}}{\frac{3}{4}}$$

Calculate the square root of 100 and get 10.

$$\frac{\frac{\sqrt{21}}{10}\times \frac{3}{2}}{\frac{2}{5}}-\frac{\left(\frac{3}{4}\right)^{2}\times \frac{4}{3}}{\frac{3}{4}}$$

Multiply \frac{\sqrt{21}}{10} times \frac{3}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\sqrt{21}\times 3}{10\times 2}}{\frac{2}{5}}-\frac{\left(\frac{3}{4}\right)^{2}\times \frac{4}{3}}{\frac{3}{4}}$$

Divide \frac{\sqrt{21}\times 3}{10\times 2} by \frac{2}{5} by multiplying \frac{\sqrt{21}\times 3}{10\times 2} by the reciprocal of \frac{2}{5}.

$$\frac{\sqrt{21}\times 3\times 5}{10\times 2\times 2}-\frac{\left(\frac{3}{4}\right)^{2}\times \frac{4}{3}}{\frac{3}{4}}$$

Cancel out 5 in both numerator and denominator.

$$\frac{3\sqrt{21}}{2\times 2\times 2}-\frac{\left(\frac{3}{4}\right)^{2}\times \frac{4}{3}}{\frac{3}{4}}$$

Multiply 2 and 2 to get 4.

$$\frac{3\sqrt{21}}{4\times 2}-\frac{\left(\frac{3}{4}\right)^{2}\times \frac{4}{3}}{\frac{3}{4}}$$

Multiply 4 and 2 to get 8.

$$\frac{3\sqrt{21}}{8}-\frac{\left(\frac{3}{4}\right)^{2}\times \frac{4}{3}}{\frac{3}{4}}$$

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

$$\frac{3\sqrt{21}}{8}-\frac{\frac{9}{16}\times \frac{4}{3}}{\frac{3}{4}}$$

Multiply \frac{9}{16} times \frac{4}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3\sqrt{21}}{8}-\frac{\frac{9\times 4}{16\times 3}}{\frac{3}{4}}$$

Do the multiplications in the fraction \frac{9\times 4}{16\times 3}.

$$\frac{3\sqrt{21}}{8}-\frac{\frac{36}{48}}{\frac{3}{4}}$$

Reduce the fraction \frac{36}{48} to lowest terms by extracting and canceling out 12.

$$\frac{3\sqrt{21}}{8}-\frac{\frac{3}{4}}{\frac{3}{4}}$$

Divide \frac{3}{4} by \frac{3}{4} to get 1.

$$\frac{3\sqrt{21}}{8}-1$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{8}{8}.

$$\frac{3\sqrt{21}}{8}-\frac{8}{8}$$

Since \frac{3\sqrt{21}}{8} and \frac{8}{8} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\sqrt{21}-8}{8}$$

Factor

$$\frac{3 \sqrt{21} - 8}{8} = 0.71846588560844$$