Exponents (Grade 9) — Mathematics Grades 8–11

The laws of exponents: multiplying and dividing powers, powers of powers, and zero and negative exponents.

Gr 9 Term 1 — ExponentsTerm 1

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What gets asked

Simplify a product or quotient of powers with the same base.
Apply a power to a power, or to every factor inside a bracket.
Evaluate an expression that contains a power of zero.
Simplify an expression that contains a negative exponent.

You must be able to

Multiply or divide powers that have the same base by adding or subtracting exponents.
Raise a power to a power, and raise every factor inside a bracket to a power.
Use the rule that any nonzero number to the power 0 equals 1.
Rewrite a negative exponent as a fraction with a positive exponent.

Traps that cost marks

Multiplying the exponents instead of adding them when the bases are the same. For the same base, multiplying means add the exponents: $2^{3} \times 2^{4} = 2^{7}$ .
Applying the power of zero to the variable only, not the whole bracket. The power 0 applies to everything inside the bracket: $(4 x^{2})^{0} = 1$ , not $4$ .
Treating a negative exponent as a negative number. A negative exponent gives a reciprocal, not a negative value: $5^{- 2} = \frac{1}{25}$ .

Worked example

Simplify:

\frac{x ^{5} \times x ^{2}}{x ^{4}}

Multiply the top: $x^{5} \times x^{2} = x^{7}$ .
Divide by the bottom: $x^{7} \div x^{4} = x^{3}$ .
The simplified answer is $x^{3}$ .

The exponent laws

Short rules that save you from writing out long strings of the same number.

base: The number or letter being multiplied. In $2^{5}$ the base is $2$ .
exponent: The small raised number. For a whole-number exponent it says how many times to multiply the base by itself.
power: A base and an exponent written together, such as $2^{5}$ .
exponential form: A number written as a power instead of as an ordinary number, as $32$ is written $2^{5}$ .

A power has two parts. In $2^{5}$ the base is $2$ and the exponent is $5$ . It means $2 \times 2 \times 2 \times 2 \times 2$ , which comes to $32$ . Writing $32$ as $2^{5}$ is putting it into exponential form. It never means $2 \times 5$ .

To multiply powers of the same base, add the exponents. $2^{3} \times 2^{4}$ is three $2$ s followed by four more $2$ s. That is seven $2$ s in a row, so the answer is $2^{7}$ .

To divide powers of the same base, subtract the exponents. $x^{7} \div x^{4} = x^{3}$ , because four of the seven $x$ s cancel with the four underneath.

A power raised to a power multiplies the exponents. A power of a product hands the exponent to every factor inside the bracket, the number included: $(3 x)^{2} = 9 x^{2}$ .

Anything except $0$ , raised to the power $0$ , is $1$ . All of these laws need the same base. $x^{3} + x^{4}$ cannot be joined at all, because adding powers is not multiplying them.

Rules to remember

$a^{m} \times a^{n} = a^{m + n}$
$(a^{m})^{n} = a^{mn}$ , and $(a t)^{n} = a^{n} t^{n}$
$a^{0} = 1$ for any $a$ except $0$

Examples

Simplify:

2^{3} \times 2^{4}

Worked answer

The bases match, so add the exponents.
$3 + 4 = 7$ , so the answer is $2^{7}$ .
As an ordinary number, $2^{7} = 128$ .

Answer: $2^{7}$ , or $128$

Multiplying the exponents would have given $2^{12}$ , which is far too big.

Simplify:

(3 x^{2})^{3}

Worked answer

The exponent $3$ goes onto everything inside the bracket.
The number too: $3^{3} = 27$ .
For the letter, multiply the exponents: $(x^{2})^{3} = x^{6}$ .
So $(3 x^{2})^{3} = 27 x^{6}$ .

Answer: $27 x^{6}$

Two laws worked together, and the $3$ in front is the part most often left alone.

Simplify:

\frac{a ^{6} \times a ^{2}}{a ^{3}} \times (5 b)^{0}

Worked answer

Top first: $a^{6} \times a^{2} = a^{8}$ .
Now divide: $a^{8} \div a^{3} = a^{5}$ .
The power $0$ covers the whole bracket: $(5 b)^{0} = 1$ .
Multiplying by $1$ changes nothing.

Answer: $a^{5}$

Three laws in one question, and the bracket disappeared instead of leaving a $5$ behind.

Traps

Multiplying the exponents instead of adding them when the bases are the same. For the same base, multiplying means add the exponents: $2^{3} \times 2^{4} = 2^{7}$ .
Applying the power of zero to the variable only, not the whole bracket. The power 0 applies to everything inside the bracket: $(4 x^{2})^{0} = 1$ , not $4$ .
Reading $2^{3}$ as $2 \times 3$ . The exponent counts the factors: $2^{3} = 2 \times 2 \times 2 = 8$ , not $6$ .
Writing $(3 x^{2})^{3}$ as $3 x^{6}$ . The number inside is raised as well: $3^{3} = 27$ , so the answer is $27 x^{6}$ .

Learn and practise “The exponent laws” in the app →

Negative exponents

A minus sign up there does not make the answer negative. It turns the number upside down.

negative exponent: An exponent below zero. It tells you to turn the power over, so $2^{- 3}$ means $\frac{1}{2 ^{3}}$ .
reciprocal: The result of dividing $1$ by a number. The reciprocal of $8$ is $\frac{1}{8}$ .
index: Another name for the exponent. The plural of index is indices, and exam papers use both words.

A negative exponent does not make a number negative. It replaces the power with its reciprocal. $2^{- 3}$ is $\frac{1}{2 ^{3}}$ , which comes to $\frac{1}{8}$ . That is small and positive.

Negative exponents come out of dividing. $\frac{2 ^{3}}{2 ^{5}} = 2^{- 2}$ , because $3 - 5 = - 2$ . The long way round gives $\frac{8}{32}$ , which is $\frac{1}{4}$ , and the two answers agree.

The index belongs only to what it sits on. In $2 x^{- 3}$ the $- 3$ sits on the $x$ alone, so the answer is $\frac{2}{x ^{3}}$ . In $(2 x)^{- 3}$ the bracket catches it, and the $2$ goes underneath as well.

The laws still need the same base, and adding is still not multiplying. $2^{3} + 2^{4}$ is $8 + 16 = 24$ , not $2^{7}$ . Work each power out first, then add them.

Powers grow fast. A message forwarded to $3$ new people each round reaches $3^{4} = 81$ people after four rounds. That is $3 \times 3 \times 3 \times 3$ , not $3 \times 4$ .

Rules to remember

$a^{- m} = \frac{1}{a ^{m}}$ for any $a$ except $0$
$5^{- 2} = \frac{1}{25}$
$2^{3} + 2^{4} = 8 + 16 = 24$

Examples

Write without a negative exponent:

3^{- 2}

Worked answer

A negative exponent asks for the reciprocal.
$3^{- 2} = \frac{1}{3 ^{2}}$ .
$3^{2} = 9$ , so the answer is $\frac{1}{9}$ .

Answer: $\frac{1}{9}$

The answer came out small and positive, which is what a negative exponent always does.

Write without a negative exponent:

4 x^{- 2}

Worked answer

The index sits on the $x$ only, not on the $4$ .
$x^{- 2} = \frac{1}{x ^{2}}$ .
So $4 x^{- 2} = 4 \times \frac{1}{x ^{2}} = \frac{4}{x ^{2}}$ .

Answer: $\frac{4}{x ^{2}}$

The $4$ stayed on top, and that is what tells this apart from $(4 x)^{- 2}$ .

Calculate:

2^{- 3} \times 2^{5} + 3^{0}

Worked answer

Same base, so add the exponents: $2^{- 3} \times 2^{5} = 2^{2}$ .
$2^{2} = 4$ .
$3^{0} = 1$ .
Add them: $4 + 1 = 5$ .

Answer: $5$

The negative exponent behaved like any other exponent once the bases matched.

Traps

Treating a negative exponent as a negative number. A negative exponent gives a reciprocal, not a negative value: $5^{- 2} = \frac{1}{25}$ .
Writing $2 x^{- 3}$ as $\frac{1}{2 x ^{3}}$ . The index sits on the $x$ alone, so $2 x^{- 3} = \frac{2}{x ^{3}}$ .
Adding the exponents when the powers are added rather than multiplied. $2^{3} + 2^{4}$ is $8 + 16 = 24$ , not $2^{7}$ .
Working $3^{4}$ out as $3 \times 4$ . Multiply the base by itself four times: $3 \times 3 \times 3 \times 3 = 81$ .

Learn and practise “Negative exponents” in the app →

Scientific notation

A short way to write the very big and the very small, so you can compare them at a glance.

scientific notation: A number written as a coefficient times a power of ten, as $4500$ is written $4, 5 \times 1 0^{3}$ .
coefficient: The number in front of the power of ten. It must be at least $1$ and less than $10$ .
ordinary form: The everyday way of writing a number, digit by digit, such as $4500$ .

Scientific notation splits a number into two parts: a coefficient, and a power of ten. Long strings of zeros disappear, and two numbers become easy to line up against each other.

For a large number, move the decimal comma left until one digit stands in front of it. Count the moves, and that count is the exponent. To return to ordinary form, move the comma back the same number of places.

For a small number, move the comma right instead, until it has passed the first digit that is not $0$ . Now the exponent is negative: $0, 00053 = 5, 3 \times 1 0^{- 4}$ . The number is small, not negative.

To multiply, multiply the coefficients and add the exponents. If the new coefficient lands on $10$ or more, put that coefficient into scientific notation too, and add the exponents again.

To compare, read the power of ten first. $2 \times 1 0^{6}$ beats $9 \times 1 0^{3}$ easily, even though $9$ is the bigger digit. The coefficient only decides when the powers of ten match.

Rules to remember

$4500 = 4, 5 \times 1 0^{3}$
$0, 00053 = 5, 3 \times 1 0^{- 4}$
$1 0^{- 4} = \frac{1}{10 000}$

Examples

Write

63000000

in scientific notation.

Worked answer

Put the comma just after the first digit: $6, 3$ .
Count how many places the comma moved: $7$ .
So $63000000 = 6, 3 \times 1 0^{7}$ .

Answer: $6, 3 \times 1 0^{7}$

The exponent counts the moves of the comma, not the zeros and not the digits.

Write

0, 00042

in scientific notation.

Worked answer

Move the comma past the first digit that is not $0$ : $4, 2$ .
It moved $4$ places to the right, so the exponent is $- 4$ .
$0, 00042 = 4, 2 \times 1 0^{- 4}$ .

Answer: $4, 2 \times 1 0^{- 4}$

The minus sign sits on the exponent, so it makes the number small and not negative.

The road from Cape Town to Johannesburg is about

1, 4 \times 1 0^{3}

km. A taxi drives it

50

times in a year. Write the year's distance in scientific notation.

Worked answer

Multiply the coefficients: $1, 4 \times 50 = 70$ .
The power of ten has not changed, so this is $70 \times 1 0^{3}$ .
But the coefficient must be under $10$ , and $70 = 7 \times 1 0^{1}$ .
Add the exponents: $1 + 3 = 4$ .
The distance is $7 \times 1 0^{4}$ km.

Answer: $7 \times 1 0^{4}$ km

Reaching $70$ was not the end of the work. The coefficient still had to come back into range.

Traps

Leaving a coefficient of $10$ or more, such as $25 \times 1 0^{1}$ . The coefficient must be at least $1$ and under $10$ . Write $250$ as $2, 5 \times 1 0^{2}$ .
Giving a positive exponent for a number smaller than $1$ . A small number takes a negative exponent: $0, 00053 = 5, 3 \times 1 0^{- 4}$ .
Adding the coefficients while the powers of ten are still different. Make the exponents match first, so $3 \times 1 0^{4} + 2 \times 1 0^{3}$ becomes $3 \times 1 0^{4} + 0, 2 \times 1 0^{4}$ .
Comparing two numbers by their coefficients only. Read the power of ten first. $2 \times 1 0^{6}$ is far bigger than $9 \times 1 0^{3}$ .

Learn and practise “Scientific notation” in the app →

Grade 9 Exponents

What gets asked

You must be able to

Traps that cost marks

Worked example

The exponent laws

Rules to remember

Examples

Traps

Negative exponents

Rules to remember

Examples

Traps

Scientific notation

Rules to remember

Examples

Traps

About this material