Q5(i):

Rationalise the denominators of the following: (i) $\frac{1}{\sqrt{7}}$

Initial Setup & Mathematical Objective

We are given the fractional expression:

$ \frac{1}{\sqrt{7}} $

The denominator of this fraction is $\sqrt{7}$, which is an irrational number. In mathematics, it is a standard convention to express fractions with a rational number in the denominator to facilitate easier addition, subtraction, and comparison of fractions. The process of converting an irrational denominator into a rational one without altering the overall value of the expression is known as rationalisation.

Step 1: Identifying the Rationalising Factor

To eliminate the square root from the denominator, we must multiply it by a value that results in a perfect square under the radical. [Per the fundamental property of radicals, $\sqrt{x} \cdot \sqrt{x} = \sqrt{x^2} = x$ for any positive real number $x$].

Given the denominator is $\sqrt{7}$, the smallest and most direct rationalising factor is $\sqrt{7}$ itself, because:

$ \sqrt{7} \times \sqrt{7} = \sqrt{49} = 7 $

Step 2: Applying the Multiplicative Identity Property

To ensure that the value of the original fraction remains unchanged, we must multiply the entire expression by $1$. We can express $1$ as a fraction where the numerator and the denominator are both equal to our rationalising factor, $\sqrt{7}$. [By the Multiplicative Identity Property, $a \cdot 1 = a$].

We set up the multiplication as follows:

$ \frac{1}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} $

Step 3: Algebraic Simplification

We now perform the multiplication across the numerators and the denominators respectively:

• Numerator Calculation: Multiply the numerators together.
  $ 1 \times \sqrt{7} = \sqrt{7} $
• Denominator Calculation: Multiply the denominators together.
  $ \sqrt{7} \times \sqrt{7} = \sqrt{7 \times 7} = \sqrt{49} = 7 $

Combining the simplified numerator and denominator yields the final rationalised expression:

$ \frac{\sqrt{7}}{7} $

Notice that the denominator is now $7$, which is a rational integer, fulfilling the objective of the problem.

Final Solution: The rationalised form of $\frac{1}{\sqrt{7}}$ is $\frac{\sqrt{7}}{7}$.