Subjects algebra

Reciprocal Expression

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1. The problem asks for the reciprocal (مقلوب) of the number inside the image, which is $\sqrt{3} - 2$. 2. The reciprocal of a number $x$ is $\frac{1}{x}$. 3. So, we want to find $\frac{1}{\sqrt{3} - 2}$. 4. To simplify this expression, we rationalize the denominator by multiplying numerator and denominator by the conjugate of the denominator, which is $\sqrt{3} + 2$. 5. This gives: $$\frac{1}{\sqrt{3} - 2} \times \frac{\sqrt{3} + 2}{\sqrt{3} + 2} = \frac{\sqrt{3} + 2}{(\sqrt{3} - 2)(\sqrt{3} + 2)}$$ 6. Using the difference of squares formula, $(a - b)(a + b) = a^2 - b^2$, the denominator becomes: $$ (\sqrt{3})^2 - (2)^2 = 3 - 4 = -1 $$ 7. So the expression simplifies to: $$ \frac{\sqrt{3} + 2}{-1} = -\sqrt{3} - 2 $$ 8. Therefore, the reciprocal of $\sqrt{3} - 2$ is $-\sqrt{3} - 2$.