Rationalizing the Denominator of 1 by √5-√3

To rationalize the denominator of 1 by √5-√3, we need to eliminate the square root in the denominator by multiplying both the numerator and denominator by a suitable expression. In this case, we will use the conjugate of the denominator, which is √5+√3, to simplify the expression.

Step 1: Understand the Conjugate

The conjugate of a binomial expression is found by changing the sign between the terms. In this case, the conjugate of √5-√3 is √5+√3.

Step 2: Multiply the Numerator and Denominator by the Conjugate

To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of the denominator (√5+√3). This is done to eliminate the square root in the denominator.

The expression becomes:
1 * (√5+√3) / (√5-√3) * (√5+√3)

Step 3: Simplify the Expression

Now, we will simplify the expression by multiplying the numerator and denominator:

Numerator: 1 * (√5+√3) = √5+√3

Denominator: (√5-√3) * (√5+√3) = (√5)^2 - (√3)^2
= 5 - 3
= 2

Therefore, the expression simplifies to (√5+√3) / 2.

Step 4: Final Result

The rationalized form of 1 / (√5-√3) is (√5+√3) / 2.

Conclusion:

By multiplying the numerator and denominator by the conjugate of the denominator (√5+√3), we were able to eliminate the square root in the denominator and rationalize the expression. The final result is (√5+√3) / 2.