To simplify the expression [tex]\(\frac{16}{\sqrt{41} - 5}\)[/tex], we need to rationalize the denominator. This process involves eliminating the square root in the denominator by multiplying the numerator and the denominator by the conjugate of the denominator. Here's the step-by-step solution:
### Step 1: Identify the Conjugate
The original expression is [tex]\(\frac{16}{\sqrt{41} - 5}\)[/tex]. The conjugate of the denominator [tex]\(\sqrt{41} - 5\)[/tex] is [tex]\(\sqrt{41} + 5\)[/tex].
### Step 2: Multiply Numerator and Denominator by the Conjugate
To rationalize the denominator, multiply both the numerator and the denominator by [tex]\(\sqrt{41} + 5\)[/tex]:
[tex]\[
\frac{16}{\sqrt{41} - 5} \times \frac{\sqrt{41} + 5}{\sqrt{41} + 5} = \frac{16(\sqrt{41} + 5)}{(\sqrt{41} - 5)(\sqrt{41} + 5)}
\][/tex]
### Step 3: Simplify the Denominator
The denominator [tex]\((\sqrt{41} - 5)(\sqrt{41} + 5)\)[/tex] is a difference of squares, which simplifies as follows:
[tex]\[
(\sqrt{41} - 5)(\sqrt{41} + 5) = (\sqrt{41})^2 - (5)^2 = 41 - 25 = 16
\][/tex]
So, the simplified denominator is 16.
### Step 4: Simplify the Numerator
The numerator becomes:
[tex]\[
16(\sqrt{41} + 5) = 16\sqrt{41} + 80
\][/tex]
### Step 5: Final Simplification
We now have:
[tex]\[
\frac{16(\sqrt{41} + 5)}{16}
\][/tex]
We can now simplify this by canceling the common factor of 16:
[tex]\[
16(\sqrt{41} + 5) \div 16 = \sqrt{41} + 5
\][/tex]
Therefore, the simplified form of the fraction [tex]\(\frac{16}{\sqrt{41} - 5}\)[/tex] is:
[tex]\[
\frac{16\sqrt{41} + 80}{16} = 11.403124237432849
\][/tex]
The final answer is 11.403124237432849.
