Rationalize the denominator and simplify to find the value of `4/(sqrt(5)+sqrt(3))`, given that
`sqrt(5)=2.236 and sqrt(3)=1.732`

Answer

To rationalize the denominator and simplify the expression \( \frac{4}{\sqrt{5} + \sqrt{3}} \), follow these steps:

### Step 1: Multiply by the Conjugate
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator. The conjugate of \( \sqrt{5} + \sqrt{3} \) is \( \sqrt{5} - \sqrt{3} \).

\[ \frac{4}{\sqrt{5} + \sqrt{3}} \cdot \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}} = \frac{4(\sqrt{5} - \sqrt{3})}{(\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3})} \]

### Step 2: Simplify the Denominator
Using the difference of squares formula \( (a + b)(a - b) = a^2 - b^2 \), we can simplify the denominator:

\[ (\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2 \]

### Step 3: Substitute the Values
Now substitute the values of \( \sqrt{5} \) and \( \sqrt{3} \) into the numerator:

\[ = \frac{4(\sqrt{5} - \sqrt{3})}{2} \]

### Step 4: Simplify the Numerator
Now, simplify the expression:

\[ = \frac{4(\sqrt{5} - \sqrt{3})}{2} = 2(\sqrt{5} - \sqrt{3}) \]

### Step 5: Substitute the Values of Square Roots
Substituting the given values \( \sqrt{5} = 2.236 \) and \( \sqrt{3} = 1.732 \):

\[ = 2(2.236 - 1.732) \]

### Step 6: Perform the Subtraction
Calculate \( 2.236 - 1.732 \):

\[ = 2(0.504) \]

### Step 7: Final Calculation
Now multiply by 2:

\[ = 1.008 \]

Thus, the value of \( \frac{4}{\sqrt{5} + \sqrt{3}} \) is \( \boxed{1.008} \).

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