Basic Algebraic Manipulation and Indices

Reading an algebraic expression

An algebraic expression combines numbers, variables and operations. Addition and subtraction separate it into terms. For example,

$5x^2-3x+7$

has the three terms $5x^2$, $-3x$ and $7$. The sign belongs to its term, so the second term is $-3x$, not merely $3x$.

A coefficient is the number multiplying a variable part. The coefficient of $5x^2$ is $5$, while the coefficient of $-x$ is $-1$. Recognising the precise variable part matters because it tells us which terms may be collected.

Collecting like terms

Like terms have identical variable parts, with the same letters raised to the same powers. They represent the same kind of quantity, just as three metres and five metres can be combined to make eight metres.

In

$4x+7x-2x,$

all three terms are multiples of $x$. Add their coefficients:

$4x+7x-2x=(4+7-2)x=9x.$

The same idea works with several variables and powers:

$6a^2b-9a^2b+4ab^2=-3a^2b+4ab^2.$

The two remaining terms are unlike because $a^2b$ and $ab^2$ have different indices. They cannot be collected.

A useful distinction is that addition does not use the multiplication law for indices. Thus

$x^2+x^2=2x^2,$

whereas

$x^2\times x^2=x^4.$

Constants can be collected with constants, but not with variable terms. For example,

$3x+8-5x-2=-2x+6.$

Multiplying a term over a bracket

A bracket groups its contents. Multiplication is distributive, so a term outside a bracket must multiply every term inside:

$a(b+c)=ab+ac.$

For example,

$3x(2x+5)=6x^2+15x.$

The first product is $3x\times2x=6x^2$, and the second is $3x\times5=15x$.

Signs need particular care. Multiplying by a negative term changes each sign:

$-4y(2y-3)=-8y^2+12y.$

This happens because $-4y\times2y=-8y^2$ and $-4y\times(-3)=+12y$.

After expanding, collect any like terms that arise. For instance,

$5(x+2)-3x=5x+10-3x=2x+10.$

Taking out common factors

Factorising reverses expansion. To factorise an expression, identify something that divides every term and place it outside a bracket.

In

$12x+18,$

both terms share the factor $6$:

$12x+18=6(2x+3).$

For an algebraic common factor, use the variables shared by every term. For example,

$15x^3y+10x^2y^2=5x^2y(3x+2y).$

Here $5$ is the highest common numerical factor. Both terms contain at least $x^2$ and at least $y$, so $5x^2y$ is a common factor. Dividing each original term by it gives the contents of the bracket.

Taking out a negative factor can sometimes make the first term inside the bracket positive:

$-6x^2+9x=-3x(2x-3).$

Expansion provides a reliable check: multiplying the outside factor back over the bracket must reproduce every original term.

What an index means

For a positive integer index, a power is repeated multiplication:

$a^4=a\times a\times a\times a.$

The base is $a$ and the index is $4$. This meaning explains the index laws rather than making them arbitrary rules.

When multiplying powers with the same base, the repeated factors join together:

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

For example, $x^3\times x^5=x^8$. Coefficients are multiplied separately, so

$3x^2\times4x^5=12x^7.$

When dividing powers with the same non-zero base, matching factors cancel:

$\frac{a^m}{a^n}=a^{m-n}\qquad(a\ne0).$

Thus $y^7\div y^3=y^4$. If the subtraction gives a negative index, then

$a^{-n}=\frac{1}{a^n}\qquad(a\ne0).$

For example, $x^2\div x^5=x^{-3}=\frac1{x^3}$.

Powers of powers and products

Raising a power to another power repeats the entire power. The indices therefore multiply:

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

For example,

$(x^3)^4=x^{12}.$

If a product is raised to a power, every factor is affected:

$(ab)^n=a^nb^n.$

This includes numerical coefficients:

$(3x^2)^3=3^3(x^2)^3=27x^6.$

By contrast, $(3x)^2$ is $9x^2$, not $3x^2$, because the whole product inside the bracket is squared.

For a non-zero base, division also gives the zero-index result:

$a^0=1,$

because $a^m\div a^m=a^{m-m}=a^0$, while any non-zero quantity divided by itself is $1$.

Simplifying expressions with several operations

Work according to the structure of the expression. Expand brackets where needed, simplify products and powers using the index laws, and then collect like terms.

For example,

$2x(x^2+3x)-x^3=2x^3+6x^2-x^3=x^3+6x^2.$

In a product such as

$2a^3b\times5a^2b^4,$

multiply the coefficients and combine matching bases:

$2a^3b\times5a^2b^4=10a^5b^5.$

There is no index rule that turns a sum such as $a^3+a^2$ into one power. Index laws describe products, quotients and powers; collecting like terms describes sums and differences. Choosing the rule from the operation shown is the central skill.

Collect like terms

Combine only terms with identical variable parts. Add or subtract their coefficients:

$7x^2-3x^2=4x^2.$

Terms such as $x$, $x^2$ and $xy$ are unlike and remain separate.

Expand one term over a bracket

Multiply every term inside:

$a(b+c)=ab+ac,$

$a(b-c)=ab-ac.$

Track signs carefully, especially when the outside factor is negative. Collect like terms after expanding if possible.

Take out common factors

Factorising reverses expansion:

$ab+ac=a(b+c).$

Choose a factor shared by every term. For complete factorisation, take out the highest common numerical factor and every common variable to its lowest occurring power. Check by expanding.

Laws of indices

For matching bases:

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

$\frac{a^m}{a^n}=a^{m-n}\quad(a\ne0)$

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

For powers of products and quotients:

$(ab)^n=a^nb^n$

$\left(\frac ab\right)^n=\frac{a^n}{b^n}\quad(b\ne0)$

Also:

$a^0=1\quad(a\ne0),\qquad a^{-n}=\frac1{a^n}\quad(a\ne0).$

Multiply numerical coefficients separately. Remember that index laws do not combine unlike bases or replace the rules for addition.