Final Answer:
The task is to rationalise the denominator, which means rewriting a fraction so that the bottom part (denominator) does not have any square roots or irrational numbers. You do this by multiplying the top and bottom by a clever form of 1 — usually the conjugate if there’s a plus or minus with a root.
For example:
- If the denominator is √2, multiply top and bottom by √2.
- If the denominator is 2 + √3, multiply top and bottom by 2 – √3 (the conjugate).
This makes the denominator a normal number without roots.
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Explanation:
Let’s say you have a fraction like 4 / (2 + √3). The bottom has a square root, which we don’t want. So we fix it.
Step 1: Find the “conjugate” of the bottom. That means flip the sign in the middle. So for (2 + √3), the conjugate is (2 – √3).
Step 2: Multiply both the top and bottom of the fraction by that conjugate. So:
4 / (2 + √3) × (2 – √3) / (2 – √3)
Step 3: Multiply the bottoms first: (2 + √3)(2 – √3) = 2² – (√3)² = 4 – 3 = 1. Nice! No more square roots.
Step 4: Multiply the tops: 4 × (2 – √3) = 8 – 4√3.
Step 5: Put it together: (8 – 4√3) / 1 = 8 – 4√3.
And that’s your answer — no square roots in the bottom!
It’s like cleaning up the messy part of the fraction so it’s easier to work with later.
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Teach the Topic:
Rationalising the denominator is just a fancy way of saying: “get rid of the square root in the bottom of a fraction.”
Why? Because fractions with square roots in the bottom are harder to use in math problems. So we clean them up.
How? We multiply the top and bottom by the same thing — but that thing is chosen carefully to cancel out the square root.
Simple example:
What is 6 / √3 ?
Multiply top and bottom by √3:
(6 × √3) / (√3 × √3) = (6√3) / 3 = 2√3.
Now the bottom is just 3 — no square root! Easy.
You’re basically making friends with the square root by giving it a partner to cancel itself out.
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Practice Task:
Rationalise the denominator of:
5 / √5
Parent Tip: Review the logic above to help your child master the concept of rationalize denominator worksheet.
