Powers, Roots & Standard Form (Edexcel IGCSE Maths): Revision notes

Section 1

What do negative and fractional powers actually mean?

Index laws extend beyond positive whole numbers. You must be able to evaluate any integer or unit-fraction power without a calculator.

Zero index: any non-zero number to the power 0 equals 1, e.g. $5^0 = 1$
Negative index: $a^{-n} = \frac{1}{a^n}$ — it means "reciprocal", not "negative number"
Unit fraction index: $a^{\frac{1}{n}} = \sqrt[n]{a}$ — the denominator is the root
General fraction index: $a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m$

Expression	Meaning	Value
$2^{-3}$	$\frac{1}{2^3}$	$\frac{1}{8}$
$9^{\frac{1}{2}}$	$\sqrt{9}$	$3$
$8^{\frac{2}{3}}$	$\left(\sqrt[3]{8}\right)^2$	$4$

Key termsindex (power)reciprocalunit fraction index

Common mistake

$a^{-n}$ does NOT make the number negative — it takes the reciprocal. $2^{-2} = \frac{1}{4}$ , not $-4$ .

Exam tip

For $a^{\frac{m}{n}}$ , always root first then power (smaller numbers to deal with): find $\sqrt[n]{a}$ , then raise the result to the power $m$ .

Section 2

How do the index laws combine powers?

When multiplying or dividing powers of the same base, combine the indices instead of expanding the numbers.

Multiplying: $a^m \times a^n = a^{m+n}$
Dividing: $a^m \div a^n = a^{m-n}$
Power of a power: $(a^m)^n = a^{mn}$
Power of a product: $(ab)^n = a^n b^n$

These laws only apply directly when the bases are identical. If bases differ, try to rewrite one as a power of the other (e.g. $4 = 2^2$ ) before combining.

Key termsbaseindex laws

Example

Simplify $\frac{2^7 \times 2^3}{2^4}$ : add the top indices then subtract, giving $2^{7+3-4} = 2^6 = 64$ .

Section 3

What does standard form represent and how do you write numbers in it?

Standard form writes very large or very small numbers compactly as: $A \times 10^{n}$ where $1 \le A < 10$ and $n$ is an integer (positive, negative, or zero).

Large numbers: $n$ is positive, e.g. $384{,}000 = 3.84 \times 10^5$
Small numbers (less than 1): $n$ is negative, e.g. $0.00021 = 2.1 \times 10^{-4}$
The power of 10 tells you how many places the decimal point moves: right for positive $n$ , left for negative $n$
Always check $A$ is between 1 and 10 — this is the most commonly lost mark

Key termsstandard formcoefficient

Common mistake

Writing $38.4 \times 10^4$ instead of $3.84 \times 10^5$ — the coefficient must be less than 10, so always adjust the power of 10 to compensate.

Section 4

How do you add, subtract, multiply and divide numbers in standard form?

Multiplying/dividing: deal with the coefficients and powers of 10 separately, then convert back to standard form if needed. $\left(A \times 10^m\right) \times \left(B \times 10^n\right) = (A \times B) \times 10^{m+n}$ $\left(A \times 10^m\right) \div \left(B \times 10^n\right) = \left(\frac{A}{B}\right) \times 10^{m-n}$

Adding/subtracting: the powers of 10 must match first — convert one number so both have the same power, then add/subtract the coefficients only.

Use a calculator's standard form (EXP or $\times 10^x$ ) key to check answers, but you must be able to do these by hand.

Key termslike powers

Example

$(3 \times 10^4) + (5 \times 10^3)$ : rewrite as $(30 \times 10^3) + (5 \times 10^3) = 35 \times 10^3 = 3.5 \times 10^4$ .

Exam tip

After multiplying or dividing, always re-check the coefficient is still between 1 and 10 — you often need to adjust the power of 10 by one at the end.

Section 5

How do surds and roots relate to powers?

Roots are the inverse of powers, and integer/fractional indices link the two topics together.

$\sqrt{a} = a^{\frac{1}{2}}$ and $\sqrt[3]{a} = a^{\frac{1}{3}}$
Square rooting a negative number has no real solution — but a negative number cubed gives a negative result, e.g. $\sqrt[3]{-8} = -2$
Perfect squares (1, 4, 9, 16, 25...) and perfect cubes (1, 8, 27, 64...) should be known from memory to evaluate roots without a calculator
Estimating roots: $\sqrt{50}$ lies between $\sqrt{49}=7$ and $\sqrt{64}=8$ , so it is close to 7.1

Key termsrootperfect squareperfect cube

Think of it like this

Powers and roots are opposite operations, like multiplying and dividing — a root "undoes" the corresponding power.

Must Know

$a^0 = 1$ and $a^{-n} = \frac{1}{a^n}$ for any non-zero $a$
$a^{\frac{1}{n}} = \sqrt[n]{a}$ and $a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m$
Index laws (same base only): $a^m \times a^n = a^{m+n}$ , $a^m \div a^n = a^{m-n}$ , $(a^m)^n = a^{mn}$
Standard form: $A \times 10^n$ , where $1 \le A < 10$ and $n \in \mathbb{Z}$
To add/subtract in standard form, match the powers of 10 first; to multiply/divide, combine coefficients and add/subtract powers separately
Always check the final coefficient is between 1 and 10, adjusting the power of 10 if not

That's the notes covered.

Carry on to the next subtopic.