Laws of Indices | New Millennium Academy | Class 7

New Millennium Academy · Birauta, Pokhara-17

Laws of Indices

Mathematics Grade 7 Chapter: Indices & Surds

Law 1Zero Index
$x^0 = 1$

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

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

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

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

Summary of 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	$\dfrac{x^m}{x^n} = x^{m-n}$

INDICES

Introduction

Law 1 — Zero Index Law

Law 2 — Product Law

Law 3 — Quotient Law

Summary of Laws