Express as an equivalent fraction with rational denominator.
`(1)/(sqrt((10))+sqrt((14))+sqrt((15))+sqrt((21)))`

Answer

To express the fraction \( \frac{1}{\sqrt{10} + \sqrt{14} + \sqrt{15} + \sqrt{21}} \) as an equivalent fraction with a rational denominator, we will follow these steps:

### Step 1: Identify the Denominator
The denominator is \( \sqrt{10} + \sqrt{14} + \sqrt{15} + \sqrt{21} \).

### Step 2: Rationalize the Denominator
To rationalize the denominator, we will multiply the numerator and the denominator by the conjugate of the denominator. The conjugate in this case is \( \sqrt{10} + \sqrt{21} - \sqrt{14} - \sqrt{15} \).

### Step 3: Multiply the Numerator and Denominator
We multiply both the numerator and the denominator by the conjugate:

\[ \frac{1 \cdot (\sqrt{10} + \sqrt{21} - \sqrt{14} - \sqrt{15})}{(\sqrt{10} + \sqrt{14} + \sqrt{15} + \sqrt{21})(\sqrt{10} + \sqrt{21} - \sqrt{14} - \sqrt{15})} \]

### Step 4: Simplify the Denominator
Now we need to simplify the denominator using the difference of squares:

\[ (\sqrt{10} + \sqrt{21})^2 - (\sqrt{14} + \sqrt{15})^2 \]

Calculating each square:

1. \( (\sqrt{10} + \sqrt{21})^2 = 10 + 21 + 2\sqrt{10 \cdot 21} = 31 + 2\sqrt{210} \)
2. \( (\sqrt{14} + \sqrt{15})^2 = 14 + 15 + 2\sqrt{14 \cdot 15} = 29 + 2\sqrt{210} \)

Now, subtract these two results:

\[ (31 + 2\sqrt{210}) - (29 + 2\sqrt{210}) = 31 - 29 = 2 \]

### Step 5: Write the Final Expression
Now we can write the fraction as:

\[ \frac{\sqrt{10} + \sqrt{21} - \sqrt{14} - \sqrt{15}}{2} \]

### Final Result
Thus, the equivalent fraction with a rational denominator is:

\[ \frac{\sqrt{10} + \sqrt{21} - \sqrt{14} - \sqrt{15}}{2} \]