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Cambridge IGCSE · International Maths Extended
Algebraic Roots & Indices
Big idea: Powers (indices) are just repeated multiplication — and once you know
the handful of rules for combining them, you can simplify almost any expression, and even solve
for an unknown that's hiding inside a power.
Summary
Index laws are shortcut rules for multiplying, dividing, and raising powers to
powers — they work identically for numbers and algebra.
Anything to the power 1 is itself; anything to the power 0 is 1.
Negative powers mean "flip it" — a⁻ⁿ = 1/aⁿ.
Fractional powers mean "root it" — a^(1/n) = ⁿ√a, and
more generally a^(m/n) = (ⁿ√a)ᵐ.
You can combine negative and fractional powers together — negative flips it,
the denominator of the fraction tells you the root, the numerator tells you the power.
To find an unknown inside a power, get both sides of the equation over the
same base, then set the powers equal to each other.
1. The Laws of Indices
Think of an index (or "power," or "exponent") as shorthand for repeated multiplication.
c³ just means "multiply c by itself 3 times": c × c × c.
Every index law below is really just a shortcut that falls straight out of that idea — if you ever
forget a rule mid-exam, you can rebuild it by writing the powers out longhand.
The power of 1 and the power of 0
Rule
a¹ = a a⁰ = 1
Anything to the power of 1 is just itself. Anything (except 0) to the power of 0 is always 1 — no matter how big or ugly the base looks.
This one trips people up because it feels like it should be 0. But think about the division rule below and it'll make sense: a³ ÷ a³ = a⁰, and anything divided by itself is 1. So a⁰ must equal 1.
Multiplying and dividing powers with the same base
Rule
aᵐ × aⁿ = aᵐ⁺ⁿ
Same base, multiplying → ADD the powers.
Rule
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Same base, dividing → SUBTRACT the powers.
Why this works
d⁵ ÷ d² = (d×d×d×d×d) ÷ (d×d)
Two of the d's on top cancel with the two on the bottom, leaving d×d×d = d³
Which matches d⁵⁻² = d³ ✓
Raising a power to another power
Rule
(aᵐ)ⁿ = aᵐⁿ
A power OF a power → MULTIPLY the powers together.
This is different from the multiplication rule above — don't mix them up! a³ × a² = a⁵ (add), but (a³)² = a⁶ (multiply). The bracket is the giveaway: brackets around a power mean "multiply the powers."
Powers of products and fractions
Rule
(ab)ⁿ = aⁿbⁿ (a/b)ⁿ = aⁿ/bⁿ
If the power is applied to a whole product or fraction, it applies to EVERY part inside separately.
Practice
Simplify: (a) 3x⁷ × 6x⁴ (b) 6x⁷ ÷ 3x⁴ (c) (3x⁷)²
2. Negative Indices — "Flip It"
A negative power doesn't mean the answer is negative — it means reciprocal
(flip the fraction upside down). It's a natural extension of the division rule: if
a² ÷ a⁵ = a⁻³, but longhand division tells you
a²÷a⁵ = 1/a³, then a⁻³ and
1/a³ must be the same thing.
Rule
a⁻¹ = 1/a a⁻ⁿ = 1/aⁿ
A negative power is the reciprocal of the positive power.
Rule
(a/b)⁻ⁿ = (b/a)ⁿ = bⁿ/aⁿ
A fraction to a negative power flips the fraction upside down, THEN raises it to the positive power.
Memory trick
Negative power = "flip me over." k⁻³ isn't "negative something" — it's
literally just 1/k³. The minus sign never survives into the final answer as an actual negative number (unless the base itself was negative).
Practice
Simplify (l/m)⁻³
3. Fractional Indices — "Root It"
This is the concept most students find hardest to trust at first, so let's build it from scratch.
We know that (a^(1/2))² = a^(1/2 × 2) = a¹ = a — using the "power of a
power" rule from Topic 1. So whatever a^(1/2) is, squaring it gives back a.
But the only thing that squares to give a is √a.
That's the whole secret: a fractional power is just a hidden root.
Unit fractional powers (1/n)
Rule
a^(1/n) = ⁿ√a
The power 1/n means "take the nth root." Power 1/2 = square root, power 1/3 = cube root, and so on.
Negative fractional powers
Rule
a^(−1/n) = 1 / a^(1/n) = 1 / ⁿ√a
Combine both tricks: the fraction tells you to root it, the minus sign tells you to flip it. Order doesn't matter — do them in whichever order feels easiest.
General fractional powers (m/n)
Rule
a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ)
The denominator (n) is the root. The numerator (m) is the power. You can do the root first then the power, OR the power first then the root — both give the same answer, but rooting first usually keeps the numbers smaller and easier to work with.
How to read a^(m/n) at a glance
Bottom number (denominator) = which root. Top number (numerator) = which power.
For s^(2/3): denominator 3 → cube root; numerator 2 → square it.
So it's "cube root of s, then squared" — or equally, "cube root of s²." Try both ways on a small number like s = 8 and check you get the same answer (you should get 4 either way).
Worked Example — rewriting a root as a fractional power
Rewrite 1 / ³√(x⁴) in the form xⁿ, where n is a negative fraction.
Step 1 — turn the cube root into a power of 1/3: 1 / (x⁴)^(1/3)
Step 2 — use (aᵐ)ⁿ = aᵐⁿ to simplify the denominator: 1 / x^(4/3)
Step 3 — use a⁻ⁿ = 1/aⁿ to bring it up top as a negative power:
Answer: x^(−4/3)
Practice
Evaluate 27^(2/3) without a calculator.
Practice
Evaluate 16^(−1/4) without a calculator.
4. Finding an Unknown Inside a Power
Sometimes the unknown, x, isn't sitting quietly on its own — it's stuck up in the exponent, like
7⁴ˣ. These equations look scary but there's one golden rule that
cracks almost all of them at IGCSE Extended level:
The golden rule
If both sides of an equation have the same base number, then their
powers must be equal too. So if 4^(3x) = 4⁹, then
straight away 3x = 9, giving x = 3.
Your whole job is usually just to force both sides onto the same base first — using the index
laws from Topics 1–3 — and then the equation collapses into ordinary algebra.
Worked Example — simplify first, then match bases
Solve: 3^(2x) × 3⁴ = 3¹⁸
Step 1 — combine the left-hand side using aᵐ × aⁿ = aᵐ⁺ⁿ: 3^(2x+4) = 3¹⁸
Step 2 — same base (3) on both sides, so the powers must match: 2x + 4 = 18
Step 3 — solve: x = 7
Worked Example — a two-unknown power equation
Find the constants m and a given that (ax⁶)^(1/m) = 8x³
Step 1 — apply the power to BOTH parts inside the bracket, using (ab)ⁿ = aⁿbⁿ: a^(1/m) × x^(6/m) = 8x³
Step 2 — the equation naturally splits into an "x part" and a "number part." Match the x-parts (same base, x): x^(6/m) = x³ → 6/m = 3 → m = 2
Step 3 — match the number parts, now substituting m = 2: a^(1/2) = 8 → √a = 8
Step 4 — square both sides: a = 64
Practice
If qˣ = (q² × q⁵) ÷ q¹⁰, find x.
Practice
Solve for x: 2^(5x) = (2³)⁴
What to Memorise
Every law used in this chapter, at a glance:
Rule
Meaning
a¹ = a
Anything to power 1 is itself
a⁰ = 1
Anything to power 0 is 1
aᵐ × aⁿ = aᵐ⁺ⁿ
Same base, multiplying → add powers
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Same base, dividing → subtract powers
(aᵐ)ⁿ = aᵐⁿ
Power of a power → multiply powers
(ab)ⁿ = aⁿbⁿ
Power applies to every factor in a product
(a/b)ⁿ = aⁿ/bⁿ
Power applies to top and bottom of a fraction
a⁻ⁿ = 1/aⁿ
Negative power = reciprocal
(a/b)⁻ⁿ = (b/a)ⁿ
Negative power on a fraction = flip, then raise
a^(1/n) = ⁿ√a
Unit fraction power = nth root
a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ)
Denominator = root, numerator = power
a^(−1/n) = 1/ⁿ√a
Negative fractional power = flip AND root
Base
The number or letter being raised to a power. In 3^x, "3" is the base.
Index / Power / Exponent
The small raised number telling you how many times to multiply the base by itself. All three words mean the same thing.
Reciprocal
"1 over" the original — flipping a fraction upside down. This is what a negative power produces.
Same-base matching
The technique of forcing both sides of an equation onto the same base number so their powers can be set equal.
Concepts Checklist
Tick each one off only once you could explain it to a friend without looking back at your notes.
Exam Tips & Common Mistakes
Mistake: adding powers when you should multiply (or vice versa)a³ × a² = a⁵ (ADD, because you're multiplying two separate power-terms), but
(a³)² = a⁶ (MULTIPLY, because it's a power OF a power). Students mix these up constantly
under exam pressure — if you see a bracket wrapped around a power, that's your multiply signal.
Mistake: thinking a negative power makes the answer negative2⁻³ is NOT −8. It's 1/2³ = 1/8, a positive fraction. The minus sign in the
exponent only ever means "flip it," never "make it negative."
Mistake: rooting AFTER raising to a big power instead of before
For a^(m/n), always try the root first if you can (ⁿ√a, then raise to power m), not the other way round.
For 27^(2/3), rooting first gives ³√27 = 3, then 3² = 9 — easy mental maths. Cubing first would give
27² = 729, then needing ³√729 — much harder without a calculator, though it gives the same final answer.
Mistake: forgetting a⁰ = 1 has to have a matching base
x⁰ = 1 for any value of x (except x = 0 itself). Don't confuse this with 0ⁿ, which is 0 for any positive n.
What examiners are actually checking
In "find x" power questions, examiners want to see the working step where you get both sides onto
the same base before writing the linear equation — jumping straight to the answer without
showing this step can lose method marks even if your final answer is correct. Always write the
"same base, so powers are equal" line explicitly.
Two-part power equations
When a question like (ax⁶)^(1/m) = 8x³ has two unknowns, split the equation into its "letter part"
(the x terms) and its "number part" (the plain numbers) and solve each separately — find m from the
x-parts first, then substitute it in to find a from the number parts.