MTH1W Powers, Zero and Negative Exponents, Scientific Notation: Notes and Practice

Take a sheet of paper about 0.1 mm thick and imagine folding it in half again and again. One fold makes it 0.2 mm thick, two folds 0.4 mm, three folds 0.8 mm: the thickness doubles every time. After 10 folds it would be $0.1 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 102.4$ mm, about the width of your hand. (Real paper gets too stiff to fold that many times, but the arithmetic is the point.) Writing ten 2s in a row is clumsy and easy to miscount. A power packs repeated multiplication into a short symbol: $2^{10}$ . Powers also let scientists write enormous numbers (the distance to the Sun) and tiny ones (the width of a blood cell) without strings of zeros.

Base, exponent and power

power 2^{5} = 5 factors of the base 2 2 \times 2 \times 2 \times 2 \times 2 = 32

In $2^{5}$ , the base is 2 (the number being multiplied) and the exponent is 5 (how many times the base appears as a factor). The whole expression $2^{5}$ is called a power, read "2 to the power of 5" or "2 to the fifth." An exponent of 2 is read "squared" and an exponent of 3 "cubed," because $5^{2}$ is the area of a 5 by 5 square and $5^{3}$ is the volume of a 5 by 5 by 5 cube. Any number to the exponent 1 is just itself: $7^{1} = 7$ . The exponent is not a multiplier: $2^{5}$ is 32, not $2 \times 5 = 10$ .

Fig 1A 3 by 3 square of 9 unit squares shows 3 squared = 9, and a 3 by 3 by 3 cube of 27 unit cubes shows 3 cubed = 27.

Quick checkTry it before you read on

Evaluate

4^{3}

ABrackets:

(- 2)^{4}

The base is $- 2$ ; the exponent applies to the negative sign too.
$(- 2) (- 2) (- 2) (- 2) = 16$
With a negative base in brackets: an even exponent gives a positive result, an odd exponent gives a negative one, e.g. $(- 2)^{3} = - 8$ .

BNo brackets:

- 2^{4}

The base is $2$ ; the exponent is done first, then the negative sign is applied.
$- (2 \times 2 \times 2 \times 2) = - 16$
Order of operations does the power before the "times $- 1$ ," so write brackets whenever you mean a negative base.

Quick checkTry it before you read on

Which of these has a positive value?

The halving pattern: zero and negative exponents

Fig 2Powers of 2 from 2 to the 3 down to 2 to the minus 3. Each step down lowers the exponent by 1 and divides the value by 2.

What could $2^{0}$ or $2^{- 1}$ mean? You cannot write "zero factors of 2" or "negative one factors." Instead, look at the pattern. Going down the list $2^{3} = 8$ , $2^{2} = 4$ , $2^{1} = 2$ , each time the exponent drops by 1, the value is divided by 2. Keep the pattern going: $2 \div 2 = 1$ , so $2^{0} = 1$ . Then $1 \div 2 = \frac{1}{2}$ , so $2^{- 1} = \frac{1}{2}$ , and $2^{- 2} = \frac{1}{4}$ , $2^{- 3} = \frac{1}{8}$ . The same thing happens with any nonzero base: for powers of 10, dividing by 10 each step gives $1 0^{0} = 1$ , $1 0^{- 1} = 0.1$ , $1 0^{- 2} = 0.01$ . A negative exponent means "one over the positive power." For a base like 2 or 10, that gives a small positive number, not a negative one.

a^{0} = 1 a^{- n} = \frac{1}{a ^{n}} (a \neq = 0)

Two traps with zero and negative exponents

First,

2^{- 3}

\frac{1}{8}

, a positive number; it is not

- 8

. Second, the base cannot be 0:

0^{- 1}

would mean

\frac{1}{0}

, and division by zero is undefined. The expression

0^{0}

is also left undefined in this course.

Quick checkTry it before you read on

Continue the pattern

3^{2} = 9

3^{1} = 3

, … What are

3^{0}

and

3^{- 1}

Quick checkTry it before you read on

Write

1 0^{- 2}

as a decimal.

VideoKhan Academy builds zero and negative exponents from a pattern: each time the exponent drops by 1, divide by the base. It is the same reasoning as the halving pattern for powers of 2 in this section.Watch on YouTube

Scientific notation

The Sun is about 150 000 000 km from Earth, and a red blood cell is about 0.000 008 m across. Counting zeros in numbers like these is slow and error-prone. Scientific notation writes a positive number as $a \times 1 0^{n}$ , where the coefficient $a$ is at least 1 and less than 10, and the exponent $n$ is an integer. The exponent counts how many places the decimal point moves: a positive exponent means a large number (move the point right to get back to standard form), and a negative exponent means a number between 0 and 1 (move the point left). So $150000000 = 1.5 \times 1 0^{8}$ and $0.000 008 = 8 \times 1 0^{- 6}$ . A negative number keeps its sign in front, as in $- 3.2 \times 1 0^{4} = - 32000$ .

Quantity	Standard form	Scientific notation
Distance from Earth to the Sun	about 150 000 000 km	$1.5 \times 1 0^{8}$ km
Population of Canada (2024)	about 41 000 000	$4.1 \times 1 0^{7}$
Speed of light	about 300 000 000 m/s	$3.0 \times 1 0^{8}$ m/s
One millimetre in metres	0.001 m	$1 \times 1 0^{- 3}$ m
Width of a red blood cell	about 0.000 008 m	$8 \times 1 0^{- 6}$ m

Very large and very small measurements in standard form and in scientific notation.

Place the decimal point just after the first nonzero digit to get the coefficient (for 47 000 000 that is 4.7; for 0.000 302 it is 3.02).
Count how many places the decimal point moved to get there.
If the original number is 10 or more, the exponent is positive; if it is between 0 and 1, the exponent is negative. (Numbers from 1 up to but not including 10 use $1 0^{0}$ .)
Check by moving the point back: $4.7 \times 1 0^{7}$ should return 47 000 000.

Fig 3Digit strips for 150 000 000 and 0.000 008. The decimal point hops 8 places left to give 1.5 × 10⁸, and 6 places right to give 8 × 10⁻⁶.

Reading a calculator

Many calculators show scientific notation with an E: 1.5E8 means

1.5 \times 1 0^{8}

and 8E-6 means

8 \times 1 0^{- 6}

. To enter

3.0 \times 1 0^{8}

, use the EE or EXP key rather than typing "times 10 to the power," which avoids order-of-operations errors.

Quick checkTry it before you read on

Which is 6 200 000 written in scientific notation?

Quick checkTry it before you read on

Write

7.1 \times 1 0^{- 3}

in standard form.

Powers, Zero and Negative Exponents, Scientific Notation

Key things to remember

Base, exponent and power

The halving pattern: zero and negative exponents

Scientific notation

Worked examples

Where marks get lost

Flashcards 8

Practice