Which of the following is greater : `(12)/(sqrt(2))` or `(18)/(sqrt(3))` ?

Answer

To determine which of the two fractions, \(\frac{12}{\sqrt{2}}\) or \(\frac{18}{\sqrt{3}}\), is greater, we can use the method of rationalization. Here are the steps to solve the problem:

### Step 1: Write down the fractions
We have:
\[ \frac{12}{\sqrt{2}} \quad \text{and} \quad \frac{18}{\sqrt{3}} \]

### Step 2: Rationalize the denominators
To compare these fractions, we will rationalize the denominators by multiplying both the numerator and denominator by the square root of the denominator.

For \(\frac{12}{\sqrt{2}}\):
\[ \frac{12}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{12\sqrt{2}}{2} = 6\sqrt{2} \]

For \(\frac{18}{\sqrt{3}}\):
\[ \frac{18}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{18\sqrt{3}}{3} = 6\sqrt{3} \]

### Step 3: Compare the results
Now we need to compare \(6\sqrt{2}\) and \(6\sqrt{3}\). Since both terms have a common factor of 6, we can compare \(\sqrt{2}\) and \(\sqrt{3}\).

### Step 4: Determine the values of \(\sqrt{2}\) and \(\sqrt{3}\)
We know that:
\[ \sqrt{2} \approx 1.414 \quad \text{and} \quad \sqrt{3} \approx 1.732 \]

Since \(1.732 > 1.414\), it follows that:
\[ \sqrt{3} > \sqrt{2} \]

### Step 5: Conclusion
Since \(6\sqrt{3} > 6\sqrt{2}\), we conclude that:
\[ \frac{18}{\sqrt{3}} > \frac{12}{\sqrt{2}} \]

Thus, the final answer is:
\[ \frac{18}{\sqrt{3}} \text{ is greater than } \frac{12}{\sqrt{2}}. \]

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