Laws of exponentsIB MYP Maths Extended: Revision notes

Section 1

Exponents and the multiplication law

An exponent (or index) tells you how many times to multiply the base by itself: $2^5=2\times2\times2\times2\times2=32$ . When you multiply powers with the same base, add the exponents: $a^m\times a^n=a^{m+n}.$ Example: $3^4\times3^2=3^6$ . With coefficients, multiply the numbers and add the exponents: $3x^2\times4x^5=12x^7$ .

Key termsexponentbase

Common mistake

Multiplying the exponents: $x^3\times x^4$ is $x^7$ , not $x^{12}$ . Add when you multiply powers.

Section 2

The division law

When you divide powers with the same base, subtract the exponents: $\frac{a^m}{a^n}=a^{m-n}.$ Example: $\dfrac{5^7}{5^3}=5^4$ . With coefficients, divide the numbers and subtract the exponents: $\dfrac{12a^5}{4a^2}=3a^3$ .

Key termsquotient

Exam tip

Always subtract the bottom exponent from the top exponent, in that order.

Section 3

Power of a power

When a power is raised to another power, multiply the exponents. Every part inside the bracket is raised to the power: $(a^m)^n=a^{mn},\qquad (ab)^n=a^nb^n.$ Example: $(x^2)^5=x^{10}$ and $(2x^3)^2=2^2x^6=4x^6$ .

Key termspower of a power

Common mistake

Forgetting the coefficient: $(2x^3)^2$ is $4x^6$ , not $2x^6$ . The 2 is squared as well.

Section 4

The zero exponent

Any non-zero number raised to the power 0 equals 1: $a^0=1\quad(a\ne0).$ Why? $a^3\div a^3=a^{3-3}=a^0$ , but anything divided by itself is 1. So $5^0=1$ , $(-8)^0=1$ and $(3x)^0=1$ .

Key termszero exponent

Common mistake

Thinking $5^0=0$ . The answer is 1.

Section 5

Negative exponents

A negative exponent means "one over" the positive power: $a^{-n}=\frac{1}{a^n}.$ So $2^{-3}=\frac{1}{2^3}=\frac18$ and $x^{-2}=\frac{1}{x^2}$ . The number does not become negative. For fractions, flip the fraction: $\left(\frac23\right)^{-2}=\left(\frac32\right)^2=\frac94$ .

Key termsreciprocalnegative exponent

Common mistake

Writing $2^{-3}=-8$ or $-6$ . A negative exponent gives a reciprocal, so $2^{-3}=\frac18$ .

Section 6

Simplifying expressions

Combine the coefficients first, then deal with each letter separately using the laws above. Finish with positive exponents only unless told otherwise. Worked example: $\dfrac{12a^5b^{-2}}{4a^2b}$ .

Coefficients: $12\div4=3$ .
$a$ : $a^{5-2}=a^3$ .
$b$ : $b^{-2-1}=b^{-3}=\frac{1}{b^3}$ . So the answer is $\dfrac{3a^3}{b^3}$ . For numbers, write them with the same base first: $\dfrac{8^2}{2^3}=\dfrac{(2^3)^2}{2^3}=\dfrac{2^6}{2^3}=2^3=8$ .

Key termssimplify

Exam tip

Dividing by a negative exponent adds: $\frac{x^3}{x^{-4}}=x^{3-(-4)}=x^7$ . Put the subtraction in brackets to avoid sign slips.

That's the notes covered.

Carry on to the next subtopic.

Exam questions on Laws of exponents

Let $x$ be any non-zero number.
Write $\dfrac{x^5\times x^{-2}}{x^{-4}}$ as a single power of $x$ .2 marks
Do not use a calculator in this question. Give each answer as an integer or a fraction in its simplest form.
Find the value of $\left(\dfrac{2}{3}\right)^{-2}$ .2 marks
Maya is simplifying algebraic expressions. All the letters represent non-zero numbers.
Simplify $\dfrac{12a^5b^{-2}}{4a^2b}$ . Write your answer with positive exponents only.3 marks

See the full worksheet

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).