Powers, Roots and Standard Form

A power (or index) tells you how many times to multiply a number by itself. A root is the inverse operation.

Expression	Meaning	Value
$5^2$	$5\times 5$	25
$2^{10}$	$2\times 2\times \cdots \times 2$ (10 times)	1024
$\sqrt{49}$	square root of 49	7
$\sqrt[3]{27}$	cube root of 27	3
$\sqrt[4]{81}$	fourth root of 81	3

Powers of 2, 3, 4 and 5 to recognise:

Base	Powers
2	$2^1=2,2^2=4,2^3=8,2^4=16,2^5=32,2^6=64,2^7=128,2^8=256,2^9=512,2^{10}=1024$
3	$3^1=3,3^2=9,3^3=27,3^4=81,3^5=243$
4	$4^2=16,4^3=64,4^4=256$
5	$5^2=25,5^3=125,5^4=625$

(Extra context — Higher tier students should also be able to estimate non-exact powers and roots, e.g. $\sqrt{50}\approx 7.07$ by reasoning $7^2=49$ and $8^2=64$, so $\sqrt{50}$ is just above 7.)

The laws of indices apply to any base $a$ (where $a\ne 0$):

Law	Rule	Example
Multiply	$a^m\times a^n=a^{m+n}$	$2^3\times 2^4=2^7=128$
Divide	$a^m÷a^n=a^{m-n}$	$3^5÷3^2=3^3=27$
Power of a power	$(a^m{)}^n=a^{mn}$	$(5^2{)}^3=5^6=15625$
Zero index	$a^0=1$	$7^0=1$
Negative index	$a^{-n}=\frac{1}{a^n}$	$4^{-2}=\frac{1}{4^2}=\frac{1}{16}$

Worked example — simplify $\frac{6x^5\times 2x^3}{4x^4}$:

Numerator: $6x^5\times 2x^3=12x^8$

Divide: $\frac{12x^8}{4x^4}=3x^{8-4}=3x^4$ ✓

The laws only apply when the bases are the same. $2^3\times 3^2$ cannot be simplified using index laws.

Fractional indices extend the index laws to roots.

$a^{\frac{1}{n}}=\sqrt[n]{a}\text{(the }n\text{th root of }a\text{)}$

$a^{\frac{m}{n}}={\left(\sqrt[n]{a}\right)}^m=\sqrt[n]{a^m}$

Worked examples:

Expression	Calculation	Value
$8^{\frac{1}{3}}$	$\sqrt[3]{8}$	2
$4^{\frac{3}{2}}$	${\left(\sqrt{4}\right)}^3=2^3$	8
$27^{\frac{2}{3}}$	${\left(\sqrt[3]{27}\right)}^2=3^2$	9
$16^{-\frac{3}{4}}$	$\frac{1}{{\left(\sqrt[4]{16}\right)}^3}=\frac{1}{2^3}$	$\frac{1}{8}$

The reliable order is: root first, then power — this keeps numbers smaller at each step.

Standard form (also called scientific notation) expresses very large or very small numbers as: $A\times 10^n\text{where }1\le A<10\text{ and }n\text{ is an integer}$

Converting to standard form:

Ordinary number	Standard form	Note
3,800,000	$3.8\times 10^6$	large number → positive $n$
0.000045	$4.5\times 10^{-5}$	small number → negative $n$
730	$7.3\times 10^2$

Converting from standard form:

• $2.4\times 10^3=2400$ (move decimal point 3 places right)
• $6.1\times 10^{-4}=0.00061$ (move decimal point 4 places left)

Worked example — write 0.000307 in standard form:

$0.000307=3.07\times 10^{-4}$ (decimal point moves 4 places right to get a number between 1 and 10) ✓

Multiplying: multiply the $A$ values, add the powers of 10; adjust if $A$ falls outside $[1,10)$.

Worked example: $(3\times 10^4)\times (4\times 10^3)$

$=12\times 10^7=1.2\times 10^8$ ✓ (adjust: $12=1.2\times 10^1$, so $10^7\times 10^1=10^8$)

Dividing: divide the $A$ values, subtract the powers of 10.

Worked example: $(8.4\times 10^6)÷(2\times 10^2)$

$=\frac{8.4}{2}\times 10^{6-2}=4.2\times 10^4$ ✓

Adding/subtracting in standard form — convert to ordinary numbers (or equalise the powers of 10 first):

Worked example: $3.2\times 10^4+4.5\times 10^3=32000+4500=36500=3.65\times 10^4$ ✓

Systematic listing means listing all possible outcomes in an organised way, ensuring nothing is missed.

Worked example — list all two-digit numbers that can be made using the digits 1, 2, 3 (no repetition):

12, 13, 21, 23, 31, 32 — six outcomes, found by fixing each digit in the tens position in turn.

The Product Rule for Counting (Higher): if there are $m$ ways to do one task and $n$ ways to do another task, the total number of ways to do both is $m\times n$.

Worked example — a menu has 4 starters, 5 main courses and 3 desserts. How many different three-course meals are possible?

$4\times 5\times 3=60$ different meals ✓

Worked example — how many different 3-digit codes can be made from digits 1–9 if repetition is allowed?

$9\times 9\times 9=729$ ✓

If no repetition: $9\times 8\times 7=504$ (one fewer choice at each stage).

1. Applying index laws to different bases

Index laws only work when multiplying or dividing powers with the same base. $2^3\times 3^2=8\times 9=72$ — it cannot be simplified to $6^5$.

2. Standard form — the value of A

$A$ must satisfy $1\le A<10$. Writing $38\times 10^5$ is not standard form; it should be $3.8\times 10^6$.

3. Negative index means negative value

$2^{-3}=\frac{1}{8}$, not $-8$. A negative index means "take the reciprocal of the positive power" — the result is a small positive number, not a negative number.

4. Fractional index — power before root

For $a^{\frac{m}{n}}$, computing the root first (then raising to the power) keeps numbers manageable. The mathematically equivalent approach of raising to the power first works but produces very large intermediate numbers.

Mistake	Correction
"$2^3\times 2^4=2^{12}$"	Add the indices: $2^3\times 2^4=2^7=128$
"$5^0=0$"	Any non-zero base to the power zero equals 1: $5^0=1$
"$3.8\times 10^6+2.1\times 10^6=5.9\times 10^{12}$"	Same power of 10: add $A$ values only, keep $10^6$; answer is $5.9\times 10^6$