What is the simplified value of [(√41 + √7) ÷ ( √41 – √7)]?

Options

A. (24 + √287)/17
B. (24 – √287)/17
C. (√41 + √7)/2
D. (√51 – √41)

Answer

To simplify the expression \(\frac{\sqrt{41} + \sqrt{7}}{\sqrt{41} - \sqrt{7}}\), we will follow these steps:

### Step 1: Rationalize the Denominator
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{41} + \sqrt{7}\).

\[ \frac{\sqrt{41} + \sqrt{7}}{\sqrt{41} - \sqrt{7}} \cdot \frac{\sqrt{41} + \sqrt{7}}{\sqrt{41} + \sqrt{7}} = \frac{(\sqrt{41} + \sqrt{7})^2}{(\sqrt{41})^2 - (\sqrt{7})^2} \]

### Step 2: Apply the Difference of Squares Formula
The denominator can be simplified using the difference of squares formula \(a^2 - b^2\).

\[ (\sqrt{41})^2 - (\sqrt{7})^2 = 41 - 7 = 34 \]

### Step 3: Expand the Numerator
Now, we expand the numerator \((\sqrt{41} + \sqrt{7})^2\) using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \).

\[ (\sqrt{41} + \sqrt{7})^2 = (\sqrt{41})^2 + 2(\sqrt{41})(\sqrt{7}) + (\sqrt{7})^2 = 41 + 2\sqrt{287} + 7 \]

### Step 4: Combine Like Terms in the Numerator
Now, combine the constant terms in the numerator.

\[ 41 + 7 + 2\sqrt{287} = 48 + 2\sqrt{287} \]

### Step 5: Write the Final Expression
Now we can write the entire expression:

\[ \frac{48 + 2\sqrt{287}}{34} \]

### Step 6: Simplify the Expression
We can simplify this expression further by dividing both terms in the numerator by 34:

\[ \frac{48}{34} + \frac{2\sqrt{287}}{34} = \frac{24}{17} + \frac{\sqrt{287}}{17} \]

### Final Answer
Thus, the simplified value of the expression is:

\[ \frac{24 + \sqrt{287}}{17} \]

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