Simplifying surds, laws of indices, rationalising denominators.
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Pick up your calculator and type 2. You get 1.41421356… — a decimal that never ends and never repeats. You cannot write it down exactly as a decimal. But you can write it exactly as 2.
That is what surds are for: exact values. A-Level exams almost always want exact answers, so you need to be fluent with surds and indices from day one.
You met these at GCSE. At A-Level, they must be second nature — especially with fractional and negative exponents.
For any base a=0 and any real exponents m and n:
| Law | Rule |
|---|---|
| Multiplication | am⋅an=am+n |
| Division | am÷an=am−n |
| Power of a power | (am)n=amn |
These three laws do all the heavy lifting. Practise them until they are automatic before moving on.
Drag the sliders to see the index laws in action:
Drag the sliders to explore how index laws work
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Now that the core three laws are secure, we can derive three more:
| Law | Rule |
|---|---|
| Power of 0 | a0=1 |
| Negative index | a−n=an1 |
| Fractional index | anm=nam=(na)m |
Watch out: a0=1 is true for every non-zero value of a. It is not “undefined” and it is not 0. If you are not sure why, notice that a3÷a3=a3−3=a0, and anything divided by itself is 1.
Answer at least 3 of 3 correctly to complete this section.
This is the single most common mistake in this topic. Compare:
x−2=x21(always positive when x=0)
−x2(always negative when x=0)
These are completely different expressions. A negative exponent means “reciprocal”, not “make it negative”.
Drag the slider to see how these two expressions behave differently
Always positive — the negative index means "one over"
Always negative — the minus is applied after squaring
The denominator of the fraction is the root, and the numerator is the power.
x21=x,x31=3x,x23=(x)3=x3
Either order works — root first then power, or power first then root. Root first is usually easier with numbers.
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A surd is a root that cannot be simplified to a rational number. For example, 2, 5, and 11 are surds. But 9=3 is not a surd — it simplifies to a whole number.
a⋅b=a⋅b
To simplify a surd, find the largest square factor inside the root.
72=36⋅2=36⋅2=62
48=16⋅3=43
Type a number to simplify its square root step by step.
The surd trap: a+b=a+b. Try it: 9+16=25=5, but 9+16=3+4=7. They are not equal. The square root does not “distribute” over addition.
You can only combine like surds, just as you combine like terms in algebra:
35+75=105
But 32+43 cannot be simplified further — they are unlike surds.
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A-Level convention is: no surds in the denominator. Rationalising removes them.
Multiply top and bottom by the surd:
35=35⋅33=353
Before learning this technique, try expanding this product:
Quick check: Expand (3+2)(3−2).
You should get 9−2=7. The surd has disappeared! This is the difference of two squares at work: (a+b)(a−b)=a2−b2. That is exactly the trick we use next.
When the denominator is a+b or a−b, multiply by the conjugate. The conjugate flips the sign of the surd term.
3+21⋅3−23−2=9−23−2=73−2
This works because of the difference of two squares: (a+b)(a−b)=a2−b2. The surd disappears from the bottom.
Remember: you must multiply both the numerator and the denominator by the conjugate. Multiplying only the bottom changes the value of the fraction.
Answer at least 3 of 4 correctly to complete this section.
Simplify 75+212−48
Break each surd into its simplest form:
75=25⋅3=53
212=24⋅3=2⋅23=43
48=16⋅3=43
Now combine like surds:
53+43−43=53
Express 3−54 in the form a+b5
Multiply top and bottom by the conjugate 3+5:
3−54⋅3+53+5=(3)2−(5)24(3+5)=412+45=3+5
So a=3 and b=1.
Answer at least 3 of 4 correctly to complete this section.
Lock in what you've learned with exam-style questions and spaced repetition.
Completing the square, discriminant, solving quadratic equations and inequalities.
Solving simultaneous equations by elimination and substitution, including one linear and one quadratic.
Graphs and equations involving the modulus function, solving modulus equations and inequalities.
Domain and range, composite functions, inverse functions and their graphs.
Equation of a straight line, gradient, midpoint, distance between two points, parallel and perpendicular lines.
nth term, common difference, sum of arithmetic series, applications.
Definition of logarithms, laws of logs, change of base formula.
First principles, differentiating x^n, gradient functions, rates of change.
Constant acceleration equations (suvat), choosing and applying the correct equation.