Multiply the fraction by 2-sqrt(3)2-sqrt(3).
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To rationalize the expression $\frac{1}{2+\sqrt{3}}$, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of $2+\sqrt{3}$ is $2-\sqrt{3}$.
Step 1: Multiply the fraction by $\frac{2-\sqrt{3}}{2-\sqrt{3}}$.
$$ \frac{1}{2+\sqrt{3}} \times \frac{2-\sqrt{3}}{2-\sqrt{3}} $$
Step 2: Perform the multiplication.
The numerator becomes $1 \times (2-\sqrt{3}) = 2-\sqrt{3}$.
The denominator is in the form $(a+b)(a-b) = a^2 - b^2$, where $a=2$ and $b=\sqrt{3}$.
$$ (2+\sqrt{3})(2-\sqrt{3}) = 2^2 - (\sqrt{3})^2 $$
Step 3: Simplify the denominator.
$$ 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 $$
Step 4: Write the simplified fraction.
$$ \frac{2-\sqrt{3}}{1} = 2-\sqrt{3} $$
The rationalized expression is $\boxed{2-\sqrt{3}}$.
