Solve for a a^2+( square root of 3)^2=2^2

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

Step 1

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$a^{2} + {(3^{\frac{1}{2}})}^{2} = 2^{2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$a^{2} + 3^{\frac{1}{2} \cdot 2} = 2^{2}$

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$a^{2} + 3^{1} = 2^{2}$

Evaluate the exponent.

$a^{2} + 3 = 2^{2}$

Step 2

Raise $2$ to the power of $2$ .

$a^{2} + 3 = 4$

Step 3

Move all terms not containing $a$ to the right side of the equation.

Subtract $3$ from both sides of the equation.

$a^{2} = 4 - 3$

Subtract $3$ from $4$ .

$a^{2} = 1$

Step 4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$a = \pm \sqrt{1}$

Step 5

Any root of $1$ is $1$ .

$a = \pm 1$

Step 6

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$a = 1$

Next, use the negative value of the $\pm$ to find the second solution.

$a = - 1$

The complete solution is the result of both the positive and negative portions of the solution.

$a = 1, - 1$