C1.7 Indices I

Learning objectives

C1.7.1Understand and use indices (positive, zero and negative integers).• Understand and use indices (positive, zero and negative integers).C1.7.2Understand and use the rules of indices• Understand and use the rules of indices. e.g. find the value of 7 –2. e.g. find the value of 2–3 × 24, (23)2, 23 ÷ 24.

In $a^n$ , $a$ is the base and $n$ is the index. A positive integer index counts repeated factors; zero and negative indices extend the same pattern so that moving one index step down always divides by the base.

Integer index	Meaning	Example
$n>0$	multiply $n$ copies of the base	$2^4=2\times2\times2\times2=16$
$0$	value is 1 when the base is non-zero	$19^0=1$
$-n$	reciprocal of the matching positive power	$2^{-4}=\frac{1}{2^4}=\frac{1}{16}=0.0625$

$a^0=1,\qquad a^{-n}= rac{1}{a^n}\quad(a e0,\ n>0)$

The sequence $2^3=8$ , $2^2=4$ , $2^1=2$ , $2^0=1$ , $2^{-1}=\frac12$ divides by 2 at every step. This explains both the zero-index and negative-index definitions rather than treating them as disconnected rules.

Brackets decide whether a negative sign belongs to the base: $(-3)^2=9$ , but $-3^2=-(3^2)=-9$ . A negative index does not make the value negative; it creates a reciprocal.

The expressions $0^0$ and $0^{-n}$ are not defined here because the negative-index rule would require division by zero. This objective uses integer indices only; fractional indices are outside C1.7 Core scope.

Index laws compress repeated multiplication. They apply when the bases match, or when one complete product, quotient or power is raised to an index; the operation tells you what to do with the indices.

Structure	Index law	Why
multiply same base	$a^m\times a^n=a^{m+n}$	join the two groups of factors
divide same base	$a^m\div a^n=a^{m-n}$	cancel matching factors; $a\ne0$
power of a power	$(a^m)^n=a^{mn}$	repeat a group of $m$ factors, $n$ times
power of a product	$(ab)^n=a^n b^n$	every repeated factor contains both $a$ and $b$
power of a quotient	$(a/b)^n=a^n/b^n$	apply the power to numerator and denominator; $b\ne0$

Expression	Apply the law	Value
$2^{-3}\times2^4$	$2^{-3+4}=2^1$	$2$
$(2^3)^2$	$2^{3\times2}=2^6$	$64$
$2^3\div2^4$	$2^{3-4}=2^{-1}$	$\frac12$

$27 imes81^2=3^3 imes(3^4)^2=3^3 imes3^8=3^{11}$

When the bases already match, equate indices after simplifying. For $5^n\div5^4=5^6$ , the quotient law gives $5^{n-4}=5^6$ , so $n-4=6$ and $n=10$ .

Do not use the addition law when bases differ, and do not multiply indices when multiplying powers: $a^m\times a^n$ adds indices, whereas $(a^m)^n$ multiplies them. Also, $(a+b)^n$ does not become $a^n+b^n$ .

Next Topic

—C1.8 Standard formRead

C1.7 Indices I

Learning objectives

Interpret positive, zero and negative integer indices

Combine powers using the index laws

Next Topic