Surds: Simplifying and Rationalising
Simplify surds by taking out square factors, add and multiply them, and clear a surd from the bottom of a fraction.
What a learner can do afterwards
- Simplify √50 to 5√2 by finding the largest square factor
- Expand a bracket containing surds and collect like terms
- Rationalise a denominator such as 6/√2
1 · Read
A surd is a root you leave exact, and simplifying means hunting the biggest square factor inside. 50 splits into 25 times 2, and root 25 is 5, so root 50 becomes 5 root 2. The same hunt turns root 72 into 6 root 2.
Brackets with surds expand like ordinary brackets, then only matching roots merge. Root 2 times (root 8 + root 2): root 2 times root 8 is root 16 = 4, and root 2 times root 2 is 2, giving 6. And (root 5 + 1) times (root 5 - 1) is 5 - 1 = 4.
Examiners refuse roots on the bottom of a fraction, so you evict them by multiplying top and bottom by the root. 6 over root 2: multiply by root 2 over root 2 to get 6 root 2 over 2, which simplifies to 3 root 2. The root migrates upstairs where it belongs.
Only matching roots combine in a sum. 2 root 3 plus 5 root 3 is 7 root 3, but root 2 plus root 3 cannot merge. Simplify every root first, then look for matches.
Split out the biggest square, merge only matches, multiply away bottom roots.
2 · Watch
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Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.