Are `(sqrt3)/(2) and (2sqrt3)/(3)` reciprocals?

Answer

To determine if \(\frac{\sqrt{3}}{2}\) and \(\frac{2\sqrt{3}}{3}\) are reciprocals, we need to follow these steps:

### Step 1: Understand the concept of reciprocals
The reciprocal of a number \(\frac{a}{b}\) is defined as \(\frac{b}{a}\). Therefore, we need to find the reciprocal of \(\frac{\sqrt{3}}{2}\).

### Step 2: Find the reciprocal of \(\frac{\sqrt{3}}{2}\)
The reciprocal of \(\frac{\sqrt{3}}{2}\) can be calculated as:
\[ \text{Reciprocal} = \frac{2}{\sqrt{3}} \]

### Step 3: Simplify the second number \(\frac{2\sqrt{3}}{3}\)
Next, we need to check if \(\frac{2\sqrt{3}}{3}\) is equal to \(\frac{2}{\sqrt{3}}\).

### Step 4: Cross-multiply to compare
To compare \(\frac{2}{\sqrt{3}}\) and \(\frac{2\sqrt{3}}{3}\), we can cross-multiply:
\[ 2 \cdot 3 \quad \text{and} \quad 2\sqrt{3} \cdot \sqrt{3} \]
This gives us:
\[ 6 \quad \text{and} \quad 2 \cdot 3 = 6 \]

### Step 5: Conclusion
Since both sides are equal, we conclude that:
\[ \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \]
Thus, \(\frac{\sqrt{3}}{2}\) and \(\frac{2\sqrt{3}}{3}\) are indeed reciprocals.

### Final Answer
Yes, \(\frac{\sqrt{3}}{2}\) and \(\frac{2\sqrt{3}}{3}\) are reciprocals.

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