Rationalise the denominator: 1/(sqrt(5) - sqrt(3)).
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Step-by-Step Solution
Step 1: Identify the conjugate
To rationalize a denominator of the form (a−b), we multiply both the numerator and the denominator by its conjugate. The conjugate of (5−3) is (5+3). This is because when we multiply a binomial by its conjugate, we use the identity (a−b)(a+b)=a2−b2, which eliminates the square roots.
Step 2: Multiply by the conjugate
We multiply the given fraction by a fraction equivalent to 1, which is 5+35+3. This operation does not change the value of the original expression but helps in rationalizing the denominator.
Step 3: Simplify the numerator
Multiplying the numerator by 1 simply results in the conjugate itself, which is (5+3).
Step 4: Simplify the denominator
Using the algebraic identity (a−b)(a+b)=a2−b2, we can simplify the denominator. Here, a=5 and b=3. So, (5)2−(3)2=5−3=2. This eliminates the square roots from the denominator.
Step 5: Write the rationalized expression
By combining the simplified numerator and denominator, we get the final rationalized expression. The denominator is now a rational number (2).