Solve {left(3/2sqrt{5/3}-sqrt{5/4}right)}^2

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

Evaluate

$$-\frac{5\sqrt{3}}{2}+5\approx 0.669872981$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{5}{3}} as the division of square roots \frac{\sqrt{5}}{\sqrt{3}}.

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

Rationalize the denominator of \frac{\sqrt{5}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$\left(\frac{3}{2}\times \frac{\sqrt{15}}{3}-\sqrt{\frac{5}{4}}\right)^{2}$$

Multiply \frac{3}{2} times \frac{\sqrt{15}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\left(\frac{3\sqrt{15}}{2\times 3}-\sqrt{\frac{5}{4}}\right)^{2}$$

Cancel out 3 in both numerator and denominator.

$$\left(\frac{\sqrt{15}}{2}-\sqrt{\frac{5}{4}}\right)^{2}$$

Rewrite the square root of the division \sqrt{\frac{5}{4}} as the division of square roots \frac{\sqrt{5}}{\sqrt{4}}.

$$\left(\frac{\sqrt{15}}{2}-\frac{\sqrt{5}}{\sqrt{4}}\right)^{2}$$

Calculate the square root of 4 and get 2.

$$\left(\frac{\sqrt{15}}{2}-\frac{\sqrt{5}}{2}\right)^{2}$$

Since \frac{\sqrt{15}}{2} and \frac{\sqrt{5}}{2} have the same denominator, subtract them by subtracting their numerators.

$$\left(\frac{\sqrt{15}-\sqrt{5}}{2}\right)^{2}$$

To raise \frac{\sqrt{15}-\sqrt{5}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(\sqrt{15}-\sqrt{5}\right)^{2}}{2^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{15}-\sqrt{5}\right)^{2}.

$$\frac{\left(\sqrt{15}\right)^{2}-2\sqrt{15}\sqrt{5}+\left(\sqrt{5}\right)^{2}}{2^{2}}$$

The square of \sqrt{15} is 15.

$$\frac{15-2\sqrt{15}\sqrt{5}+\left(\sqrt{5}\right)^{2}}{2^{2}}$$

Factor 15=5\times 3. Rewrite the square root of the product \sqrt{5\times 3} as the product of square roots \sqrt{5}\sqrt{3}.

$$\frac{15-2\sqrt{5}\sqrt{3}\sqrt{5}+\left(\sqrt{5}\right)^{2}}{2^{2}}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$\frac{15-2\times 5\sqrt{3}+\left(\sqrt{5}\right)^{2}}{2^{2}}$$

Multiply -2 and 5 to get -10.

$$\frac{15-10\sqrt{3}+\left(\sqrt{5}\right)^{2}}{2^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{15-10\sqrt{3}+5}{2^{2}}$$

Add 15 and 5 to get 20.

$$\frac{20-10\sqrt{3}}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{20-10\sqrt{3}}{4}$$

Expand

$$-\frac{5 \sqrt{3}}{2} + 5 = 0.669872981$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{5}{3}} as the division of square roots \frac{\sqrt{5}}{\sqrt{3}}.

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

Rationalize the denominator of \frac{\sqrt{5}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$\left(\frac{3}{2}\times \frac{\sqrt{15}}{3}-\sqrt{\frac{5}{4}}\right)^{2}$$

Multiply \frac{3}{2} times \frac{\sqrt{15}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\left(\frac{3\sqrt{15}}{2\times 3}-\sqrt{\frac{5}{4}}\right)^{2}$$

Cancel out 3 in both numerator and denominator.

$$\left(\frac{\sqrt{15}}{2}-\sqrt{\frac{5}{4}}\right)^{2}$$

Rewrite the square root of the division \sqrt{\frac{5}{4}} as the division of square roots \frac{\sqrt{5}}{\sqrt{4}}.

$$\left(\frac{\sqrt{15}}{2}-\frac{\sqrt{5}}{\sqrt{4}}\right)^{2}$$

Calculate the square root of 4 and get 2.

$$\left(\frac{\sqrt{15}}{2}-\frac{\sqrt{5}}{2}\right)^{2}$$

Since \frac{\sqrt{15}}{2} and \frac{\sqrt{5}}{2} have the same denominator, subtract them by subtracting their numerators.

$$\left(\frac{\sqrt{15}-\sqrt{5}}{2}\right)^{2}$$

To raise \frac{\sqrt{15}-\sqrt{5}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(\sqrt{15}-\sqrt{5}\right)^{2}}{2^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{15}-\sqrt{5}\right)^{2}.

$$\frac{\left(\sqrt{15}\right)^{2}-2\sqrt{15}\sqrt{5}+\left(\sqrt{5}\right)^{2}}{2^{2}}$$

The square of \sqrt{15} is 15.

$$\frac{15-2\sqrt{15}\sqrt{5}+\left(\sqrt{5}\right)^{2}}{2^{2}}$$

Factor 15=5\times 3. Rewrite the square root of the product \sqrt{5\times 3} as the product of square roots \sqrt{5}\sqrt{3}.

$$\frac{15-2\sqrt{5}\sqrt{3}\sqrt{5}+\left(\sqrt{5}\right)^{2}}{2^{2}}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$\frac{15-2\times 5\sqrt{3}+\left(\sqrt{5}\right)^{2}}{2^{2}}$$

Multiply -2 and 5 to get -10.

$$\frac{15-10\sqrt{3}+\left(\sqrt{5}\right)^{2}}{2^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{15-10\sqrt{3}+5}{2^{2}}$$

Add 15 and 5 to get 20.

$$\frac{20-10\sqrt{3}}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{20-10\sqrt{3}}{4}$$