Index Laws with Letters

Using the index laws with letters: multiplying, dividing, a bracket to a power with numbers in front, and finding a missing power.

In the exam for Edexcel IGCSE Maths 4MA1, Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024 · 97 real past-paper questions below

اقرأ هذا الدرس مشروحاً بالعربي ←

What a Power Means

A power counts copies. $a^{5}$ means five a's multiplied together.

a^{5} = a \times a \times a \times a \times a

So when the exam writes $d \times d \times d \times d$ , the simplest answer is $d^{4}$ .

Rule

The small number is the index (or power). It only belongs to what is right next to it: $3 a^{2}$ is $3 \times a \times a$ .

Key idea

Every rule in this lesson comes from counting copies. If you forget a rule, write the copies out.

Multiplying and Dividing Powers

Most 1-mark questions use one of these two laws. They only work when the letter (the base) is the same.

Formula

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

$a$ is the letter, and $m$ and $n$ are its two powers. Multiplying adds the powers.

Worked example

1
$a^{2} \times a^{3}$ — The question. Both powers belong to the same letter a, so the law works.
2
$= (a \times a) \times (a \times a \times a)$ — Write out the copies: a² is 2 copies of a and a³ is 3 copies.
3
$= a \times a \times a \times a \times a$ — Drop the brackets. Everything is multiplied, so the copies join into one row.
4
$= a^{5}$ — Count them: 5 copies. That is 2 + 3, which is why multiplying adds the powers.

Formula

a^{m} \div a^{n} = a^{m - n}

Dividing takes away the powers: the top power minus the bottom power.

Worked example

1
$a^{5} \div a^{2}$ — The question. Same letter again, so the law works.
2
$= \frac{a \times a \times a \times a \times a}{a \times a}$ — Write it as a fraction: 5 copies on top, 2 on the bottom.
3
$= \frac{a \times a \times a \times a \times a}{a \times a}$ — Each a on the bottom cancels one a on top, because a ÷ a = 1.
4
$= a \times a \times a$ — 3 copies are left on top, and only 1 is left on the bottom.
5
$= a^{3}$ — 3 is 5 − 2, which is why dividing subtracts the powers.

Key idea

Divide a power by itself: $a^{5} \div a^{5} = a^{0}$ , and it is also 1. So anything to the power 0 is 1, even $(p + 3)^{0}$ .

A Bracket to a Power

This is the biggest question type, often worth 2 marks. The power outside the bracket reaches everything inside it.

Formula

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

A power of a power: multiply the two powers. $(x^{2})^{3}$ is three copies of $x^{2}$ , which is $x^{6}$ .

Formula

(ab)^{n} = a^{n} b^{n}

$a$ and $b$ are the parts multiplied inside the bracket, numbers or letters. Each part gets the power $n$ .

Worked example

1
$(3 x^{2} y^{4})^{3}$ — The question. Inside the bracket are three parts: the number 3, $x^{2}$ and $y^{4}$ .
2
$= 3^{3} \times (x^{2})^{3} \times (y^{4})^{3}$ — Give the power 3 to every part, because the power reaches everything inside.
3
$3^{3} = 3 \times 3 \times 3 = 27$ — Cube the number: three 3s multiplied. It is not 3 × 3.
4
$(x^{2})^{3} = x^{6}$ — Power of a power: multiply, 2 × 3 = 6.
5
$(y^{4})^{3} = y^{12}$ — The same law for y: 4 × 3 = 12.
6
$= 27 x^{6} y^{12}$ — Put the parts together as one term: the number first, then the letters.

Common mistake

Writing 9 for the number. The 3 is cubed too: $3^{3} = 27$ , not $3 \times 3$ .

Key idea

Show each part. The marks check the number and each letter on their own, so two parts right often still earns 1 of the 2 marks.

Multiplying Terms with Numbers

Exams often ask you to multiply two terms that each have a number in front. Deal with the numbers and each letter on its own.

Worked example

1
$4 x^{2} y^{3} \times 6 x^{5} y$ — The question. Each term has a number, some x's and some y's.
2
$= (4 \times 6) \times (x^{2} \times x^{5}) \times (y^{3} \times y)$ — Group like with like: numbers, x's, y's. Everything is multiplied, so any order is fine.
3
$4 \times 6 = 24$ — The numbers in front are ordinary numbers, so multiply them.
4
$x^{2} \times x^{5} = x^{7}$ — Same letter multiplied: add the powers, 2 + 5 = 7.
5
$y^{3} \times y^{1} = y^{4}$ — y on its own means $y^{1}$ , so 3 + 1 = 4.
6
$= 24 x^{7} y^{4}$ — Write it as one term: the number first, then the letters.

Common mistake

Adding the numbers: 4 + 6 = 10. Only the powers add. The numbers in front multiply: 4 × 6 = 24.

Key idea

Each part earns credit: a right number with one wrong power usually still gets 1 of the 2 marks. So finish every part.

Tidy the Bracket First

Sometimes the bracket holds two terms multiplied together. Tidy the inside into one term first, then use the power.

Worked example

1
$(2 x^{3} \times 5 x y^{2})^{2}$ — The question. Two terms multiply inside the bracket, and the power 2 is outside.
2
$2 \times 5 = 10$ — Tidy inside first. Start with the numbers: multiply them.
3
$x^{3} \times x^{1} = x^{4}$ — Then the x's: x on its own is $x^{1}$ , so 3 + 1 = 4. The $y^{2}$ has no partner, so it stays.
4
$(10 x^{4} y^{2})^{2}$ — Now the bracket is one term, like the earlier card.
5
$= 1 0^{2} \times (x^{4})^{2} \times (y^{2})^{2}$ — Give the power 2 to every part.
6
$1 0^{2} = 100$ — Square the number: 10 × 10.
7
$(x^{4})^{2} = x^{8}, (y^{2})^{2} = y^{4}$ — Power of a power for each letter: 4 × 2 = 8 and 2 × 2 = 4.
8
$= 100 x^{8} y^{4}$ — Put the parts together as one term.

Key idea

Write the tidied bracket down as its own line. It can earn a mark by itself, even if a later line slips.

Dividing Numbers and Letters

For a fraction with numbers and letters, deal with each part on its own.

Worked example

1
$\frac{24 x ^{5} y ^{3}}{6 x ^{2} y}$ — The question. A number and two letters on top, the same kinds on the bottom.
2
$= \frac{24}{6} \times \frac{x ^{5}}{x ^{2}} \times \frac{y ^{3}}{y}$ — Split it into small fractions: numbers, x's, y's. You can, because everything is multiplied.
3
$24 \div 6 = 4$ — The numbers are ordinary numbers, so divide them.
4
$x^{5} \div x^{2} = x^{3}$ — Same letter divided: subtract the powers, 5 − 2 = 3.
5
$y^{3} \div y^{1} = y^{2}$ — y on its own means $y^{1}$ , so 3 − 1 = 2.
6
$= 4 x^{3} y^{2}$ — Put the parts together as one term.

Common mistake

Taking away the numbers: 24 − 6. Numbers divide. Only the powers subtract.

Key idea

If the bigger power is on the bottom, that letter stays on the bottom: $\frac{x ^{2}}{x ^{5}} = \frac{1}{x ^{3}}$ .

A Fraction to a Power

If the bracket holds a fraction, the power reaches the top and the bottom.

Formula

(\frac{a}{b})^{n} = \frac{a ^{n}}{b ^{n}}

$a$ is the top, $b$ is the bottom (not 0), and $n$ is the power. Power the top, then power the bottom.

Worked example

1
$(\frac{2 x}{y ^{3}})^{4}$ — The question. The power 4 sits outside a fraction.
2
$= \frac{( 2 x ) ^{4}}{( y ^{3} ) ^{4}}$ — Give the power to the top and to the bottom.
3
$(2 x)^{4} = 2^{4} x^{4}$ — Top: the power reaches both the 2 and the x.
4
$2^{4} = 2 \times 2 \times 2 \times 2 = 16$ — Four 2s multiplied make 16, not 2 × 4 = 8.
5
$(y^{3})^{4} = y^{12}$ — Bottom: power of a power, so multiply, 3 × 4 = 12.
6
$= \frac{16 x ^{4}}{y ^{12}}$ — Top over bottom. Nothing cancels, because x and y are different letters.

Common mistake

Powering only the top and writing $\frac{16 x ^{4}}{y ^{3}}$ . The power reaches the bottom too.

Find the Missing Power

Some questions give the answer and hide a power, n. The powers follow the laws, so write an equation in the powers.

Worked example

1
$\frac{y ^{7} \times y ^{n}}{y ^{4}} = y^{9}$ — The question. Every term is a power of y, and one power is hidden.
2
$7 + n - 4 = 9$ — The powers must match on both sides. Multiplying adds n, dividing takes away 4.
3
$n + 3 = 9$ — Tidy the left side: 7 − 4 = 3.
4
$n = 6$ — Take 3 from both sides, as in any equation.
5
$7 + 6 - 4 = 9$ — Check: put n = 6 back into the powers. It gives 9, so n = 6 is right.

Rule

Same letter on both sides means the powers are equal.

Which Method? Read the Question

Look at the sign between the powers.

You see	You do
$d \times d \times d$	Count the copies: d³
$a^{3} \times a^{4}$	Add the powers
$a^{5} \div a^{2}$	Subtract the powers
$(3 x^{2} y^{4})^{3}$	Power the number; multiply each letter's power
$(2 x^{3} \times 5 x y^{2})^{2}$	Tidy inside the bracket first, then power it
$(\frac{2 x}{y ^{3}})^{4}$	Power the top and the bottom
Numbers in front	Multiply or divide the numbers as normal
$a^{0}$	It is 1
A missing power n	Write an equation in the powers

Key idea

The laws only work when the letters match. $a^{3} \times b^{2}$ does not simplify.

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