Sure, let's rationalize the denominator of the given expression step by step. The expression we need to rationalize is:
[tex]\[
\frac{3 \sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}
\][/tex]
### Step 1: Identify the Conjugate
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(\sqrt{5} - \sqrt{3}\)[/tex] is [tex]\(\sqrt{5} + \sqrt{3}\)[/tex].
### Step 2: Multiply by the Conjugate
Now multiply both the numerator and the denominator by the conjugate:
[tex]\[
\frac{3 \sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} \times \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}}
\][/tex]
This gives us:
[tex]\[
\frac{(3 \sqrt{5} + \sqrt{3})(\sqrt{5} + \sqrt{3})}{(\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})}
\][/tex]
### Step 3: Expand the Numerator
Let's expand the numerator:
[tex]\[
(3 \sqrt{5} + \sqrt{3})(\sqrt{5} + \sqrt{3})
\][/tex]
Use the distributive property (FOIL method):
[tex]\[
= (3 \sqrt{5})(\sqrt{5}) + (3 \sqrt{5})(\sqrt{3}) + (\sqrt{3})(\sqrt{5}) + (\sqrt{3})(\sqrt{3})
\][/tex]
Simplify the terms:
[tex]\[
= 3(5) + 3 \sqrt{15} + \sqrt{15} + 3
\][/tex]
Combine like terms:
[tex]\[
= 15 + 3 + 3\sqrt{15} + \sqrt{15} = 18 + 4\sqrt{15}
\][/tex]
### Step 4: Simplify the Denominator
The denominator is a difference of squares:
[tex]\[
(\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3}) = \sqrt{5}^2 - \sqrt{3}^2
\][/tex]
Simplify:
[tex]\[
= 5 - 3 = 2
\][/tex]
### Step 5: Write the Final Expression
Now we write the rationalized expression:
[tex]\[
\frac{18 + 4\sqrt{15}}{2}
\][/tex]
### Step 6: Simplify the Final Fraction
Divide each term in the numerator by the denominator:
[tex]\[
\frac{18}{2} + \frac{4\sqrt{15}}{2} = 9 + 2\sqrt{15}
\][/tex]
### Final Answer
So, the rationalized form of the given expression is:
[tex]\[
9 + 2\sqrt{15}
\][/tex]
This is the simplified result after rationalizing the denominator.
