`(sqrt(31) - sqrt(29))/(sqrt(31) + sqrt(29))` equals:

Options

A. `60 - 2sqrt(899)`
B. `30 - sqrt(899)`
C. `30 + sqrt(899)`
D. `1/(30 - sqrt(899))`

Answer

To solve the expression \((\sqrt{31} - \sqrt{29}) / (\sqrt{31} + \sqrt{29})\), we can simplify it using the technique of multiplying by the conjugate. Here’s a step-by-step solution:

### Step 1: Multiply by the Conjugate
We multiply the numerator and the denominator by the conjugate of the denominator, which is \((\sqrt{31} - \sqrt{29})\).

\[ \frac{\sqrt{31} - \sqrt{29}}{\sqrt{31} + \sqrt{29}} \cdot \frac{\sqrt{31} - \sqrt{29}}{\sqrt{31} - \sqrt{29}} = \frac{(\sqrt{31} - \sqrt{29})^2}{(\sqrt{31} + \sqrt{29})(\sqrt{31} - \sqrt{29})} \]

### Step 2: Simplify the Numerator
Now, we simplify the numerator using the identity \((a - b)^2 = a^2 - 2ab + b^2\):

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

Combining the constants:

\[ 31 + 29 - 2\sqrt{31 \cdot 29} = 60 - 2\sqrt{899} \]

### Step 3: Simplify the Denominator
Now, we simplify the denominator using the difference of squares identity:

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

### Step 4: Combine the Results
Now we can combine the results from the numerator and the denominator:

\[ \frac{60 - 2\sqrt{899}}{2} \]

### Step 5: Simplify the Fraction
We can simplify the fraction by dividing each term in the numerator by 2:

\[ 30 - \sqrt{899} \]

### Final Answer
Thus, the expression \((\sqrt{31} - \sqrt{29}) / (\sqrt{31} + \sqrt{29})\) simplifies to:

\[ 30 - \sqrt{899} \]

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