Rationalise the denominator by multiplying the numerator and denominator by the conjugate of the denominator.
This languages problem is solved step by step below, with detailed explanations to help you understand the method and arrive at the correct answer.
ScanSolve AI Answer
Step 1: Rationalise the denominator by multiplying the numerator and denominator by the conjugate of the denominator.
The conjugate of $2-\sqrt{3}$ is $2+\sqrt{3}$.
$$ \frac{2+\sqrt{3}}{2-\sqrt{3}} = \frac{2+\sqrt{3}}{2-\sqrt{3}} \times \frac{2+\sqrt{3}}{2+\sqrt{3}} $$
Step 2: Expand the numerator and the denominator.
Numerator: $(2+\sqrt{3})(2+\sqrt{3}) = 2(2) + 2\sqrt{3} + 2\sqrt{3} + (\sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3}$
Denominator: $(2-\sqrt{3})(2+\sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1$
Step 3: Simplify the expression.
$$ \frac{7 + 4\sqrt{3}}{1} = 7 + 4\sqrt{3} $$
The simplified expression is $\boxed{7 + 4\sqrt{3}}$.
Step 4: Evaluate the expression using $\sqrt{3} = 1.732$.
Substitute the value of $\sqrt{3}$ into the simplified expression $7 + 4\sqrt{3}$.
$$ 7 + 4(1.732) $$
Step 5: Perform the multiplication.
$$ 4 \times 1.732 = 6.928 $$
Step 6: Perform the addition.
$$ 7 + 6.928 = 13.928 $$
The evaluated value is $\boxed{13.928}$.
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