E2.4 Indices II

Learning objectives

E2.4.1Understand and use indices (positive, zero, negative and fractional).• Understand and use indices (positive, zero, negative and fractional).E2.4.2Understand and use the rules of indices• Understand and use positive, zero, negative and fractional indices. Solve simple exponential equations and apply the laws of indices to simplify expressions. Knowledge of logarithms is not required.

An index tells you how a base is used. Positive whole-number indices represent repeated multiplication; zero, negative and fractional indices extend the same pattern consistently.

Index form	Meaning	Condition
$a^n$	multiply $n$ copies of $a$	$n$ is a positive integer
$a^0$	$1$	$a
e0$
$a^{-n}$	$1/a^n$	$a
e0$
$a^{1/n}$	$\sqrt[n]{a}$	for real even roots, $a\ge0$
$a^{m/n}$	$\sqrt[n]{a^m}=(\sqrt[n]{a})^m$	use a real root where defined

$64^{ rac23}=\left(\sqrt[3]{64} ight)^2=4^2=16$

A negative index does not make the value negative: $5^{-2}=1/5^2=1/25$ . It moves a non-zero factor across the fraction line and changes the sign of its index.

For real $x$ , $(64x^4)^{1/2}=8x^2$ because $x^4=(x^2)^2$ and $x^2\ge0$ . Keep root conditions in mind when the variable power is not automatically non-negative.

$a^0=1$ applies only when $a
e0 $;$ 0^0 $is not assigned this value here. Also,$ a^{-n} $means a reciprocal, not$ -a^n$.

Index laws combine powers only when their bases and operation fit the law. They also let an exponential equation be solved by rewriting both sides with one common base.

Structure	Law
same base multiplied	$a^m a^n=a^{m+n}$
same base divided	$a^m/a^n=a^{m-n}$ , $a
e0$
power raised to a power	$(a^m)^n=a^{mn}$
product raised to a power	$(ab)^n=a^n b^n$
quotient raised to a power	$(a/b)^n=a^n/b^n$ , $b
e0$

Apply the outer index to every factor: $(27x^9)^{2/3}=27^{2/3}x^{9(2/3)}=9x^6$ . The coefficient and variable power are both affected.

$4^{x+1}=8^{x-1}\quad\Longrightarrow\quad2^{2x+2}=2^{3x-3}$

Equal positive bases give equal exponents, so $2x+2=3x-3$ and $x=5$ . This method needs no logarithms; first look for a common base such as $2$ , $3$ , $5$ or a reciprocal power.

Do not add indices when terms are added: $a^m+a^n$ is not generally $a^{m+n}$ . In $(a^m)^n$ , multiply the indices; in $a^m a^n$ , add them.

Next Topic

—E2.5 EquationsRead

E2.4 Indices II

Learning objectives

Interpret positive, zero, negative and fractional indices

Apply index laws and solve simple exponential equations

Next Topic