2.1 Use of symbols

Learning objectives

2.1A Symbols in equations, expressions and formulaeunderstand that symbols may be used to represent numbers in equations or variables in expressions and formulae 2.1B Generalised arithmetic rules in algebraUnderstand that algebraic expressions follow the generalised rules of arithmetic.2.1C Integer index notationuse index notation for positive and negative integer powers, including zero 2.1D Index lawsuse index laws in simple cases: xᵐ × xⁿ = xᵐ⁺ⁿ, xᵐ ÷ xⁿ = xᵐ⁻ⁿ and (xᵐ)ⁿ = xᵐⁿ 2.1HA Index notation involving fractional, negative and zerouse index notation involving fractional, negative and zero powers

A symbol can stand for an unknown number in an equation or a variable quantity in an expression or formula. Its meaning comes from the statement and context.

Algebraic object	Example	Role of symbol
expression	$3x+5$	$x$ may vary
equation	$3x+5=20$	find value(s) of $x$ making it true
formula	$A=\pi r^2$	relates area $A$ and radius $r$

Translate operations in their stated order. ‘Add 7 to $x$ , then divide by 5’ is $(x+7)/5$ , not $x+7/5$ .

Substitution replaces a symbol by a value while preserving brackets: if $x=-2$ , then $x^2=(-2)^2=4$ .

A letter is not a label that can be ignored. The same symbol has the same value throughout one expression or equation unless explicitly redefined.

Algebra follows the same commutative, associative and distributive rules as arithmetic. Symbols may be manipulated because they represent numbers.

Rule	Algebraic form	Example
commutative	$a+b=b+a$ , $ab=ba$	$4k imes2y=8ky$
associative	$(ab)c=a(bc)$	$(2x)(3y)=6xy$
distributive	$a(b+c)=ab+ac$	$3(x+4)=3x+12$

Only like terms combine by addition or subtraction: $3x+5x=8x$ , but $3x+5y$ cannot be simplified to $8xy$ .

Multiplication signs are usually omitted between a number and letters; write $8ky$ , with numerical coefficient first and letters in a consistent order.

Addition is not multiplication: $x+x=2x$ , whereas $x imes x=x^2$ .

In $x^n$ , the index records repeated multiplication when $n$ is positive. Consistent extension of the index pattern defines zero and negative powers.

Form	Meaning, where defined
$x^4$	$x imes x imes x imes x$
$x^1$	$x$
$x^0$	$1$ , for $x
e0$
$x^{-n}$	$1/x^n$ , for $x
e0$

Moving one step down in the index divides by the base: $x^3,x^2,x^1,x^0,x^{-1}$ gives $x^3,x^2,x,1,1/x$ .

$2^{-3}=1/2^3=1/8$ . A negative index creates a reciprocal; it does not make the value negative.

$x^0=1$ requires $x
e0 $, and$ x^{-n} $is undefined at$ x=0$.

Index laws compress repeated factors. They apply to powers with the same base, subject to any non-zero conditions from division.

Operation	Law	Example
multiply same base	$x^m x^n=x^{m+n}$	$y^5y^3=y^8$
divide same base	$x^m/x^n=x^{m-n}$	$a^7/a^2=a^5$
power of a power	$(x^m)^n=x^{mn}$	$(p^3)^4=p^{12}$

Handle numerical coefficients separately: $(6x^5)/(2x^2)=3x^3$ .

A subtraction producing a negative index can be rewritten reciprocally: $x^2/x^5=x^{-3}=1/x^3$ .

Do not add indices when adding powers: $x^2+x^3$ does not equal $x^5$ . Indices add only when multiplying the same base.

Fractional indices represent roots, negative indices represent reciprocals, and zero indices give 1. These meanings work together with the index laws.

Form	Equivalent	Example
$x^{1/n}$	$\sqrt[n]{x}$	$16^{1/2}=4$
$x^{m/n}$	$(\sqrt[n]{x})^m$	$8^{2/3}=4$
$x^{-m/n}$	$1/x^{m/n}$	$16^{-1/2}=1/4$

$(16x^8y^6)^{1/2}=16^{1/2}x^{8/2}y^{6/2}=4x^4y^3$ under the usual real-domain assumptions.

For real values, an even root requires a non-negative radicand; a negative power also requires a non-zero base.

$x^{m/n}$ does not mean $x^m/x^n$ . The denominator of the index names a root and the numerator names a power.

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—2.2 Algebraic manipulationRead

2.1 Use of symbols

Learning objectives

Interpret symbols in algebra

Apply arithmetic rules to algebra

Use zero and negative integer indices

Use the index laws

Use fractional, negative and zero powers

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