Algebra Toolkit Notes

Edexcel IGCSE Maths: Revision notes

Key facts

An index tells you how many times the base is multiplied by itself; $a^0=1$ and $a^{-n}=\dfrac{1}{a^n}$ .
Same base: multiplying adds indices, dividing subtracts them, a power of a power multiplies them.
$a^{\frac{m}{n}}=\left(\sqrt[n]{a}\right)^m$ : the denominator is the root.
Substitute values in brackets, especially negatives, then follow BIDMAS.
Only like terms (same letters, same powers) can be added or subtracted.

Index notation

An index says how many times the base is multiplied by itself.

$a^n$ means $a$ multiplied by itself $n$ times, so $3^4=3\times3\times3\times3=81$ and $x^5=x\times x\times x\times x\times x$ . The number being multiplied is the base.

Any non-zero number to the power 0 is 1, so $5^0=1$ and $x^0=1$ . Note $a^{\frac{1}{n}}=\sqrt[n]{a}$ .

$a^1$ $a$
$a^0$ $1$
$a^{-n}$ $\dfrac{1}{a^n}$

Worked example

Write $2^{-3}$ as a fraction and find $9^{\frac{1}{2}}$ .

1
A negative index means "one over": $2^{-3}=\dfrac{1}{2^3}=\dfrac{1}{8}$ .
2
A fractional index $\frac{1}{2}$ means square root: $9^{\frac{1}{2}}=\sqrt{9}=3$ .

What is the value of $7^0$ ?

Index laws

With the same base, add indices to multiply, subtract to divide and multiply for a power of a power.

These laws only work when the bases are the same. Terms with different bases, such as $x^3\times y^2$ , cannot be combined.

Say it as you go: "same base, multiplying, so I add the powers."

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

	Rule	Example
Multiply	$a^m\times a^n=a^{m+n}$	$x^3\times x^2=x^5$
Divide	$a^m\div a^n=a^{m-n}$	$\dfrac{a^7}{a^3}=a^4$
Power of a power	$(a^m)^n=a^{mn}$	$(x^2)^5=x^{10}$
Negative index	$a^{-n}=\dfrac{1}{a^n}$	$x^{-2}=\dfrac{1}{x^2}$
Fractional index	$a^{\frac{m}{n}}=\left(\sqrt[n]{a}\right)^m$	$8^{\frac{2}{3}}=2^2=4$

Rule

Multiply:: $a^m\times a^n=a^{m+n}$
Divide:: $a^m\div a^n=a^{m-n}$
Power of a power:: $(a^m)^n=a^{mn}$
Negative index:: $a^{-n}=\dfrac{1}{a^n}$
Fractional index:: $a^{\frac{m}{n}}=\left(\sqrt[n]{a}\right)^m$

Example

Multiply:: $x^3\times x^2=x^5$
Divide:: $\dfrac{a^7}{a^3}=a^4$
Power of a power:: $(x^2)^5=x^{10}$
Negative index:: $x^{-2}=\dfrac{1}{x^2}$
Fractional index:: $8^{\frac{2}{3}}=2^2=4$

Worked example

Simplify $\dfrac{x^5\times x^3}{x^4}$ and evaluate $8^{\frac{2}{3}}$ .

1
Multiply the top: $x^5\times x^3=x^8$ .
2
Divide: $x^8\div x^4=x^4$ .
3
$8^{\frac{2}{3}}=\left(\sqrt[3]{8}\right)^2=2^2=4$ .

Simplify $(y^3)^4$ .

Substitution

Replace each letter with its value in brackets, then follow BIDMAS.

Substitution means replacing letters with given numbers, then evaluating.

Brackets matter most with negative numbers, so you do not lose a minus sign or misapply an index. $(-3)^2=9$ , not $-9$ .

1
Write out
the expression
2
Replace
each letter with its value in brackets
3
BIDMAS
brackets, indices, then multiply/divide, then add/subtract

Substituting

Worked example

Find $2x^2-3y$ when $x=-3$ and $y=4$ .

1
Substitute in brackets: $2(-3)^2-3(4)$ .
2
Indices first: $(-3)^2=9$ , so $2\times9-12$ .
3
$18-12=6$ .

Find $x^2-5x$ when $x=-2$ .

Collecting like terms

Add or subtract the coefficients of terms with exactly the same letters and powers.

Like terms have the same letters raised to the same powers: $3x+5x=8x$ and $7y^2-2y^2=5y^2$ . But $3x$ and $5x^2$ differ in power, and $3x$ and $3y$ differ in letter.

Group the like terms, combine their coefficients, and write the answer in descending powers: $4x^2+3x-2x^2+7-5x=2x^2-2x+7$ .

Like terms

$3x$ and $5x$
$7y^2$ and $2y^2$

Not like terms

$3x$ and $5x^2$
$3x$ and $3y$

Worked example

Simplify $6a+3b-2a+5b-4$ .

1
Group the $a$ terms: $6a-2a=4a$ .
2
Group the $b$ terms: $3b+5b=8b$ .
3
The number stays: $-4$ .

Simplify $3x+5x^2-x$ .

Try an exam question

(a) Simplify $(y^3)^4$ . (b) Find the value of $x^2-5x$ when $x=-2$ . (c) Simplify $3x+5x^2-x$ .

[5 marks]

That's the notes covered.

Carry on to the next subtopic.