E1.7 Indices I

Learning objectives

E1.7.1Understand and use indices (positive, zero, negative, and fractional).• Understand and use indices (positive, zero, negative, and fractional).E1.7.2Understand and use the rules of indices• Understand and use the rules of indices. Examples include: • 662 = • 16 1644 = • find the value of 7 –2, 812, 8 3 2-. e.g. find the value of 2–3 × 24, (23)2, 23 ÷ 24.

An index tells how a base is powered. Positive whole-number indices repeat multiplication; zero, negative and fractional indices extend the same pattern so powers can also represent 1, reciprocals and roots.

Index	Meaning	Conditions	Example
$a^n$	multiply $n$ factors of $a$	$n$ positive integer	$3^4=81$
$a^0$	1	$a\ne0$	$7^0=1$
$a^{-n}$	reciprocal $1/a^n$	$a\ne0$	$2^{-4}=1/16=0.0625$
$a^{1/n}$	principal $n$ th root	real even roots need $a\ge0$	$64^{1/3}=4$
$a^{m/n}$	take the $n$ th root and raise to power $m$	same root condition	$64^{2/3}=4^2=16$

$a^{-m/n}=\frac{1}{a^{m/n}}=\frac{1}{(\sqrt[n]{a})^m}$

For $27^{-2/3}$ , take the cube root first: $27^{1/3}=3$ . Square to get $27^{2/3}=9$ , then use the negative index as a reciprocal: $27^{-2/3}=1/9$ .

The radical or a fractional power gives its principal value, not a plus-or-minus pair: $16^{1/2}=4$ . The two solutions of the equation $x^2=16$ are $x=\pm4$ , but that is a separate equation-solving statement. A negative index does not make the value negative; it makes a reciprocal.

Index laws preserve repeated multiplication. They apply when powers share the same base; rewrite numbers to a common base before combining their indices.

Structure	Index law	Condition
multiply same base	$a^m\times a^n=a^{m+n}$	same base
divide same base	$a^m\div a^n=a^{m-n}$	same non-zero base
power of a power	$(a^m)^n=a^{mn}$	multiply the indices
power of a product	$(ab)^n=a^n b^n$	power applies to every factor

Expression	Index step	Result
$2^{-3}\times2^4$	$2^{-3+4}$	$2$
$(2^3)^2$	$2^{3\times2}$	$64$
$2^3\div2^4$	$2^{3-4}=2^{-1}$	$1/2$

To write $243\times27^{2n}$ as one power of 3, first rewrite both bases: $243=3^5$ and $27^{2n}=(3^3)^{2n}=3^{6n}$ . Then multiply equal bases: $3^5\times3^{6n}=3^{6n+5}$ .

Do not add indices when adding powers: $2^3+2^4=8+16=24$ , not $2^7$ . In $(a^m)^n$ , multiply the indices; do not raise $m$ to the power $n$ . Multi-variable algebraic simplification is developed later in E2.4.

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—E1.8 Standard formRead

E1.7 Indices I

Learning objectives

Interpret positive, zero, negative and fractional indices

Combine powers using the index laws

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