Solve (a/2sqrt{3})^2+(a/2)^2=a^2

$$( \frac { a } { 2 } \sqrt { 3 } ) ^ { 2 } + ( \frac { a } { 2 } ) ^ { 2 } = a ^ { 2 }$$

Solve for a (complex solution)

$$a\in \mathrm{C}$$

Short Solution Steps

Express \frac{a}{2}\sqrt{3} as a single fraction.

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

To raise \frac{a\sqrt{3}}{2} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

Since \frac{\left(a\sqrt{3}\right)^{2}}{2^{2}} and \frac{a^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

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

Expand \left(a\sqrt{3}\right)^{2}.

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

The square of \sqrt{3} is 3.

$$\frac{a^{2}\times 3+a^{2}}{2^{2}}=a^{2}$$

Combine a^{2}\times 3 and a^{2} to get 4a^{2}.

$$\frac{4a^{2}}{2^{2}}=a^{2}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{4a^{2}}{4}=a^{2}$$

Cancel out 4 and 4.

$$a^{2}=a^{2}$$

Subtract a^{2} from both sides.

$$a^{2}-a^{2}=0$$

Combine a^{2} and -a^{2} to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any a.

$$a\in \mathrm{C}$$

Solve for a

$$a\in \mathrm{R}$$

Short Solution Steps

Express \frac{a}{2}\sqrt{3} as a single fraction.

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

To raise \frac{a\sqrt{3}}{2} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

Since \frac{\left(a\sqrt{3}\right)^{2}}{2^{2}} and \frac{a^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

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

Expand \left(a\sqrt{3}\right)^{2}.

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

The square of \sqrt{3} is 3.

$$\frac{a^{2}\times 3+a^{2}}{2^{2}}=a^{2}$$

Combine a^{2}\times 3 and a^{2} to get 4a^{2}.

$$\frac{4a^{2}}{2^{2}}=a^{2}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{4a^{2}}{4}=a^{2}$$

Cancel out 4 and 4.

$$a^{2}=a^{2}$$

Subtract a^{2} from both sides.

$$a^{2}-a^{2}=0$$

Combine a^{2} and -a^{2} to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any a.

$$a\in \mathrm{R}$$