Laws of Indices | New Millennium Academy

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Laws of Indices

Mathematics Grade 8 Chapter: Indices & Surds

Law 1Zero Index
$x^0 = 1$

Law 2Product
$a^m \cdot a^n = a^{m+n}$

Law 3Quotient
$\dfrac{x^m}{x^n} = x^{m-n}$

Law 4Distributive
$(ab)^n = a^n b^n$

Law 5Power
$(a^m)^n = a^{mn}$

Law 6Negative
$x^{-m} = \dfrac{1}{x^m}$

Introduction

What is an Index?

Let us select the number 81 and prime-factorise it.

Therefore: $\quad 81 = 3 \times 3 \times 3 \times 3$ (3 multiplied with itself 4 times)

$$81 = 3^4$$

In $3^4$: the 4 is the power / index (Pl: Indices), and the 3 is the base.

Other Examples

$2^6$ = Base 2 multiplied with itself 6 times $= 2\times2\times2\times2\times2\times2$
$5^3$ = Base 5 multiplied with itself 3 times $= 5\times5\times5$

Law 1 — Zero Index Law

Activity — Discovering the Zero Index

Observe the pattern below (subtract the index by 1 each time and divide the value by 3):

$3^4$	$= 3 \times 3 \times 3 \times 3$	$= 81$
$3^3$	$= 3 \times 3 \times 3$	$= 27$
$3^2$	$= 3 \times 3$	$= 9$
$3^1$	$= 3$	$= 3$
$3^0$	(following the pattern)	$= 1$

Continuing the pattern — subtracting 1 from the index and dividing by 3 each step — we get $3^0 = 1$.

Law 1

Zero Index Law

$$x^0 = 1$$

(where $x \neq 0$)

Examples

$(2x)^0 = 1$
$5^0 = 1$
$(2x - y)^0 = 1$
$(-5y)^0 = 1$
$9{,}999{,}999{,}999^0 = 1$

Law 2 — Product Law

Activity — Multiplying Powers with the Same Base

Let us multiply 16 and 8:

$16 \times 8 = 128$

or,$(2\times2\times2\times2)\times(2\times2\times2) = 2\times2\times2\times2\times2\times2\times2$

or,$2^4 \times 2^3 = 2^7$

or,$2^4 \times 2^3 = 2^{4+3} \quad [4+3=7]$

Conclusion

$\therefore\quad 2^4 \times 2^3 = 2^7$

This law of indices is called the Product Law.

Law 2

Product Law

$$a^m \times a^n = a^{m+n}$$

Examples

$a^3 \times a^2 = a^{3+2} = a^5$
$(2n+1)^3 \times (2n+1)^5 = (2n+1)^{3+5} = (2n+1)^8$
$(-2n)^4 \times (-2n)^2 = (-2n)^{4+2} = (-2n)^6$
$5^3 \times 5^4 \times 5^5 = 5^{3+4+5} = 5^{12}$
$\left(\dfrac{a}{b}\right)^2 \times \left(\dfrac{a}{b}\right)^3 = \left(\dfrac{a}{b}\right)^{2+3} = \left(\dfrac{a}{b}\right)^5$

Law 3 — Quotient Law

Activity I — Dividing Powers with the Same Base

Let us divide 64 by 16:

$\dfrac{64}{16} = 4$

or,$\dfrac{2\times2\times2\times2\times2\times2}{2\times2\times2\times2} = 2\times2$

or,$\dfrac{2^6}{2^4} = 2^2$

or,$\dfrac{2^6}{2^4} = 2^{6-4} \quad [6-4=2]$

Activity II — Dividing Powers with the Same Base

Let us divide 81 by 27:

$\dfrac{81}{27} = 3^1$

or,$\dfrac{3\times3\times3\times3}{3\times3\times3} = 3^1$

or,$\dfrac{3^4}{3^3} = 3^1$

or,$\dfrac{3^4}{3^3} = 3^{4-3} \quad [4-3=1]$

Conclusion

$$\frac{2^6}{2^4} = 2^{6-4} = 2^2 \qquad \frac{3^4}{3^3} = 3^{4-3} = 3^1$$

This law of indices is called the Quotient Law.

Law 3

Quotient Law

$$\frac{x^m}{x^n} = x^{m-n}$$

Examples

$\dfrac{5^6}{5^2} = 5^{6-2} = 5^4$
$\dfrac{(2a)^6}{(2a)^3} = (2a)^{6-3} = (2a)^3$
$\dfrac{(2x-y)^7}{(2x-y)^3} = (2x-y)^{7-3} = (2x-y)^4$

Law 4 — Distributive Law

Activity — Proof of Distributive Law

We have:

$(ab)^4$

$=$$(ab)(ab)(ab)(ab)$

$=$$(a \cdot a \cdot a \cdot a)(b \cdot b \cdot b \cdot b)$

$=$$a^4 \cdot b^4$

$\therefore\quad (ab)^4 = a^4 \cdot b^4$

In abstract:

$(ab)^n = (ab)(ab)(ab)\cdots \text{ n times}$

$=$$(\underbrace{a\cdot a\cdots a}_{n})\;(\underbrace{b\cdot b\cdots b}_{n})$

$=$$a^n \cdot b^n$

$\therefore\quad (ab)^n = a^n b^n$

Similarly for fractions:

$\left(\dfrac{a}{b}\right)^n = \left(\dfrac{a}{b}\right)\!\left(\dfrac{a}{b}\right)\cdots \text{ n times}$

$=$$\dfrac{a\cdot a\cdots a \text{ (n times)}}{b\cdot b\cdots b \text{ (n times)}}$

$=$$\dfrac{a^n}{b^n}$

Law 4

Distributive Law

$$(ab)^n = a^n b^n \qquad \text{and} \qquad \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$

Examples

$(xy)^3 = x^3 y^3$
$(5x)^3 = 5^3 x^3 = 125x^3$
$\left(\dfrac{x}{y}\right)^4 = \dfrac{x^4}{y^4}$

Important Note

$(x + y)^4 \neq x^4 + y^4$

The distributive law applies only to multiplication and division, NOT to addition or subtraction inside the bracket.

Law 5 — Power Law

Activity — Power of a Power

Consider $(a^3)^4$:

$= a^3 \times a^3 \times a^3 \times a^3$

$=$$a^{3+3+3+3}$

$=$$a^{12}$

$=$$a^{3 \times 4}$

In abstract:

$(a^m)^n = \underbrace{a^m \times a^m \times a^m \cdots}_{n \text{ times}}$

$=$$a^{\underbrace{m+m+m+\cdots}_{n \text{ times}}}$

$=$$a^{m \times n} = a^{mn}$

This law of index is the Power Law.

Law 5

Power Law

$$(a^m)^n = a^{mn}$$

Examples

$(a^2)^5 = a^{2\times5} = a^{10}$
$(x^3)^4 = x^{12}$

Law 6 — Negative Index Law

Activity — Discovering Negative Indices

We have $2^4 = 16$. Subtract the power by 1 and divide the value by 2 continuously:

$2^4$	$= 16$
$2^3$	$= 8$
$2^2$	$= 4$
$2^1$	$= 2$
$2^0$	$= 1$
$2^{-1}$	$= \dfrac{1}{2}$
$2^{-2}$	$= \dfrac{1}{4}$
$2^{-3}$	$= \dfrac{1}{8}$

Conclusion

Negative power does not mean a negative value.

It means dividing repeatedly.

Derivation using Quotient Law

$\dfrac{a^2}{a^5} = \dfrac{a\cdot a}{a\cdot a\cdot a\cdot a\cdot a}$

or,$a^{2-5} = \dfrac{1}{a\cdot a\cdot a} \qquad \left[\because\; \dfrac{x^m}{x^n} = x^{m-n}\right]$

or,$a^{-3} = \dfrac{1}{a^3}$

This is the Law of Negative Index.

Law 6

Negative Index Law

$$x^{-m} = \frac{1}{x^m} \qquad \text{or equivalently,} \qquad \frac{1}{x^{-m}} = x^m$$

Examples

$2^{-5} = \dfrac{1}{2^5} = \dfrac{1}{32}$
$\dfrac{1}{2^{-3}} = 2^3 = 8$

Note

Positive power is a result of repeated multiplication.

Negative power is a result of repeated division.

Summary of All Laws

Law	Name	Formula
1	Zero Index	$x^0 = 1 \;\;(x\neq0)$
2	Product Law	$a^m \times a^n = a^{m+n}$
3	Quotient Law	$\dfrac{x^m}{x^n} = x^{m-n}$
4	Distributive Law	$(ab)^n = a^n b^n \;\;$ and $\;\; \left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$
5	Power Law	$(a^m)^n = a^{mn}$
6	Negative Index	$x^{-m} = \dfrac{1}{x^m}$