Rationalize the denominator of the expression (1)/(1-sqrt(3)) and leave the answer in surd form.

Question

ScanSolve AI Tutor · Accepted Answer

To leave the answer in surd form, we need to rationalize the denominator.

Write the given expression.
$\frac{1}{1-\sqrt{3}}$
Multiply the numerator and the denominator by the conjugate of the denominator.
The conjugate of $1-\sqrt{3}$ is $1+\sqrt{3}$ .
$\frac{1}{1-\sqrt{3}} \times \frac{1+\sqrt{3}}{1+\sqrt{3}}$
Perform the multiplication.
Use the difference of squares formula $(a-b)(a+b) = a^2 - b^2$ for the denominator.
$\frac{1 \times (1+\sqrt{3})}{(1-\sqrt{3})(1+\sqrt{3})} = \frac{1+\sqrt{3}}{1^2 - (\sqrt{3})^2}$
Simplify the denominator.
$\frac{1+\sqrt{3}}{1 - 3} = \frac{1+\sqrt{3}}{-2}$
Rewrite the expression with the negative sign in the numerator or in front of the fraction.
$-\frac{1+\sqrt{3}}{2} \quad \text{or} \quad \frac{-1-\sqrt{3}}{2}$
The simplified expression in surd form is $-\frac{1+\sqrt{3}}{2}$ .

Answer