Exponents (Grade 8) — Mathematics Grades 8–11

The laws of exponents, scientific notation, and roots and powers of numbers.

Gr 8 Term 1 — ExponentsTerm 1

Practise Exponents in the app →

What gets asked

Apply an exponent law to simplify a power expression.
Write a large number in scientific notation.
Work out a square, cube, square root or cube root.
Solve a simple equation with an unknown exponent.

You must be able to

Use the product, quotient and power laws for exponents with the same base.
Convert a large number to scientific notation and back again.
Work out the square, cube, square root or cube root of a whole number or fraction.
Apply the rule that a non-zero number to the power of 0 is 1.

Traps that cost marks

Multiplying the bases as well as adding the exponents, e.g. treating $3^{4} \times 3^{2}$ as $9^{6}$ . Keep the base the same and only add the exponents: $3^{4} \times 3^{2} = 3^{6}$ .
Adding the exponents instead of multiplying them for a power of a power. For $(a^{m})^{n}$ , multiply the exponents: $(3^{2})^{4} = 3^{8}$ , not $3^{6}$ .
Writing a coefficient outside the 1-to-10 range in scientific notation, like $34 \times 1 0^{7}$ . The first factor must be between 1 and 10. Write 340 000 000 as $3, 4 \times 1 0^{8}$ .
Saying any number to the power 0 equals 0. A non-zero number to the power 0 is 1, not 0. So $9^{0} = 1$ .

Worked example

Simplify:

2^{4} \times 2^{3} \div 2^{5}

Multiply the powers on top first: $2^{4} \times 2^{3} = 2^{7}$ .
Now divide: $2^{7} \div 2^{5} = 2^{2}$ .
$2^{2} = 4$ .

Squares, cubes and roots

Squares, cubes and the roots that undo them, and the order you have to work them out in.

square: To square a number is to multiply it by itself. $7^{2} = 7 \times 7 = 49$ .
cube: To cube a number is to use it three times in one multiplication. $4^{3} = 4 \times 4 \times 4 = 64$ .
square root: The number that multiplies by itself to give it. $49 = 7$ .
cube root: The number used three times in a multiplication to give it. $364 = 4$ .
rational number: Any number you can write as one whole number over another, such as $\frac{3}{4}$ .

The small raised number tells you how many of them to multiply together. It does not tell you what to multiply by. So $7^{2}$ is $7 \times 7$ and not $7 \times 2$ . In the same way $4^{3}$ is $4 \times 4 \times 4$ , which is $64$ .

Roots go back the other way. Since $12 \times 12 = 144$ , the square root of $144$ is $12$ . Since $6 \times 6 \times 6 = 216$ , the cube root of $216$ is $6$ . Learn the squares up to $1 2^{2}$ and the cubes up to $6^{3}$ , and their roots come free.

Brackets come first, then powers and roots, and only then adding and subtracting. So $36 - 5^{2} = 6 - 25 = - 19$ . In $(7 - 4)^{3}$ you do the subtraction first, then cube the answer. Under a root sign it is the same: $16 + 9 = 25 = 5$ . Never take the root of each part on its own.

Rational numbers are squared and cubed too. Square the top and the bottom: $(\frac{3}{4})^{2} = \frac{9}{16}$ . With decimals, watch the places. One place squared gives two: $0, 2 \times 0, 2 = 0, 04$ .

Rules to remember

$1 2^{2} = 144$ and $144 = 12$
$6^{3} = 216$ and $3216 = 6$
Brackets first, then powers and roots

Examples

Work out

81 + 3^{2}

Worked answer

$81 = 9$ , because $9 \times 9 = 81$ .
$3^{2} = 3 \times 3 = 9$ .
Now add the two: $9 + 9 = 18$ .

Answer: $18$

Each part was finished on its own before anything was added.

Work out

(7 - 4)^{3}

Worked answer

The bracket comes first: $7 - 4 = 3$ .
Now cube that $3$ : $3 \times 3 \times 3 = 27$ .
So $(7 - 4)^{3} = 27$ .

Answer: $27$

The exponent sits on the whole bracket, so the subtraction had to happen first.

Work out

(\frac{3}{4})^{2}

and then

(0, 2)^{3}

Worked answer

Square the top: $3 \times 3 = 9$ .
Square the bottom: $4 \times 4 = 16$ , which gives $\frac{9}{16}$ .
For the decimal, $(0, 2)^{3} = 0, 008$ .
One decimal place cubed gives three decimal places.

Answer: $\frac{9}{16}$ and $0, 008$

Both halves of the fraction get squared, and decimal places multiply up.

Traps

Mixing squaring up with square rooting, so the square of $9$ is given as $3$ . The square of $9$ is $81$ . The square root of $9$ is $3$ . They are opposite jobs.
Giving $4^{3}$ as $12$ by multiplying the base by the small raised number. The raised number counts the factors: $4^{3} = 4 \times 4 \times 4 = 64$ .
Rooting each part on its own, so $16 + 9$ is turned into $4 + 3$ . Add inside the root first: $16 + 9 = 25 = 5$ .
Squaring only the top, so $(\frac{2}{3})^{2}$ is given as $\frac{4}{3}$ . The bottom is squared too: $(\frac{2}{3})^{2} = \frac{4}{9}$ .

Learn and practise “Squares, cubes and roots” in the app →

Exponential form

Naming the parts of a power, working out its sign, and writing huge numbers the short way.

base: The number that is being multiplied over and over. 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 written with an exponent, such as $2^{5}$ . Its value is $32$ .
scientific notation: A big number written as a number that is at least $1$ and under $10$ , times a power of $10$ .

$2^{5}$ means $2 \times 2 \times 2 \times 2 \times 2 = 32$ . The base says what to multiply and the exponent counts how many there are. It never means $2 \times 5$ . An exponent of $1$ changes nothing, so $m^{1} = m$ .

The sign of the answer depends on whether the exponent is even or odd. Minus signs cancel in pairs. An even exponent on a negative base gives a positive answer: $(- 5)^{2} = 25$ . An odd exponent leaves one minus over: $(- 2)^{5} = - 32$ .

To put powers in order, work each one out first. $2^{5} = 32$ and $5^{2} = 25$ , so $2^{5}$ is the bigger of the two, even though its base is smaller. Comparing only the bases, or only the exponents, will let you down.

Scientific notation makes a huge number short. Write the digits as a number that is at least $1$ and under $10$ . Then count how many places the comma has to move to build the number back up. That count is the exponent on the $10$ .

Rules to remember

$m^{1} = m$
$(- 2)^{4} = 16$ and $(- 2)^{5} = - 32$
$340000000 = 3, 4 \times 1 0^{8}$

Examples

Write

7 \times 7 \times 7 \times 7

as a power, then find its value

Worked answer

There are four $7$ s, so the base is $7$ and the exponent is $4$ .
In exponential form that is $7^{4}$ .
$7 \times 7 = 49$ , and $49 \times 49 = 2401$ .
So $7^{4} = 2401$ .

Answer: $7^{4} = 2401$

Counting the $7$ s gave the exponent. Nothing was multiplied by $4$ .

Work out

(- 3)^{4}

and

(- 3)^{3}

Worked answer

The exponent $4$ is even, so the answer is positive.
$(- 3)^{4} = 81$ .
The exponent $3$ is odd, so one minus sign is left over.
$(- 3)^{3} = - 27$ .

Answer: $81$ and $- 27$

The minus signs cancelled in pairs, and an odd exponent has one left over.

Write

58000000

in scientific notation

Worked answer

Put the comma just after the first digit, giving $5, 8$ .
Count the places from there to the end of the number. There are $7$ .
So the power of ten is $1 0^{7}$ .
That gives $58000000 = 5, 8 \times 1 0^{7}$ .

Answer: $5, 8 \times 1 0^{7}$

The first factor sits between $1$ and $10$ , which is what scientific notation asks for.

Traps

Reading $1 2^{2}$ as $12 \times 2$ and answering $24$ . The exponent counts the factors, so $1 2^{2} = 12 \times 12 = 144$ .
Giving $(- 5)^{2}$ as $- 25$ because the base has a minus sign. Two minus signs multiply to a plus, so $(- 5)^{2} = 25$ .
Saying $5^{2}$ beats $2^{5}$ because $5$ is the bigger base. Work both out. $5^{2} = 25$ and $2^{5} = 32$ , so $2^{5}$ is bigger.
Writing a coefficient outside the 1-to-10 range in scientific notation, like $34 \times 1 0^{7}$ . The first factor must be between 1 and 10. Write 340 000 000 as $3, 4 \times 1 0^{8}$ .

Learn and practise “Exponential form” in the app →

Laws of exponents

Short rules that let you multiply and divide powers without writing them out in full.

like bases: Two powers have like bases when the base is the same. $3^{4}$ and $3^{2}$ have like bases.
expand: To write a power out in full as a repeated multiplication. Expanding $3^{2}$ gives $3 \times 3$ .
law of exponents: A rule about powers that holds for every base and every exponent, not just for the one you tested.

Every law of exponents comes out of expanding. $3^{4} \times 3^{2}$ is four $3$ s times two more $3$ s, so it is six $3$ s in a row. That gives $3^{4} \times 3^{2} = 3^{6}$ . Keep the base and add the exponents. It only works for like bases.

Dividing cancels instead of adding. In $\frac{2 ^{7}}{2 ^{5}}$ five of the seven $2$ s cancel out, leaving two behind: $\frac{2 ^{7}}{2 ^{5}} = 2^{2} = 4$ . So you subtract the exponents. The bases must be alike here too, so $\frac{2 ^{7}}{3 ^{5}}$ cannot be shortened at all.

A power raised to a power multiplies the exponents: $(3^{2})^{4} = 3^{8}$ . Every factor inside a bracket also takes the outside exponent. So $(2 y)^{3} = 8 y^{3}$ , because the $2$ is cubed as well as the $y$ .

Divide a power by itself and the answer is $1$ . Written with the rule, $\frac{5 ^{3}}{5 ^{3}} = 5^{0}$ , and so $5^{0} = 1$ . That works for every base except $0$ .

Try a law on a second pair of numbers before you trust it. Read each one both ways as well. Whenever it helps, $3^{6}$ may be written back as $3^{4} \times 3^{2}$ . In a word problem, handle the front numbers and the powers of ten separately.

Rules to remember

$a^{m} \times a^{n} = a^{m + n}$
$\frac{a ^{m}}{a ^{n}} = a^{m - n}$
$a^{0} = 1$ for every base except $0$

Examples

Simplify:

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

Worked answer

The bases are alike, so the base stays $2$ .
Add the exponents: $4 + 3 = 7$ .
So $2^{4} \times 2^{3} = 2^{7}$ .

Answer: $2^{7}$

Nothing happened to the base. Only the exponents were added.

Simplify:

\frac{( 3 ^{2} ) ^{4}}{3 ^{5}}

Worked answer

Do the bracket first and multiply those exponents: $(3^{2})^{4} = 3^{8}$ .
Now divide, which subtracts: $\frac{3 ^{8}}{3 ^{5}} = 3^{3}$ .
Finally $3^{3} = 27$ .

Answer: $3^{3}$ , which is $27$

One law multiplied the exponents and the next one subtracted them.

A town has

6 \times 1 0^{4}

people. Each uses about

2 \times 1 0^{2}

litres of water a day. How many litres a day is that?

Worked answer

Multiply the front numbers: $6 \times 2 = 12$ .
Multiply the powers of ten: $1 0^{4} \times 1 0^{2} = 1 0^{6}$ .
So far that is $12 \times 1 0^{6}$ .
The first factor must be under $10$ , so write $1, 2 \times 1 0^{7}$ .

Answer: $1, 2 \times 1 0^{7}$ litres

The front numbers were multiplied and the exponents added, then the answer was tidied into scientific notation.

Traps

Multiplying the bases as well as adding the exponents, e.g. treating $3^{4} \times 3^{2}$ as $9^{6}$ . Keep the base the same and only add the exponents: $3^{4} \times 3^{2} = 3^{6}$ .
Adding the exponents instead of multiplying them for a power of a power. For $(a^{m})^{n}$ , multiply the exponents: $(3^{2})^{4} = 3^{8}$ , not $3^{6}$ .
Saying any number to the power 0 equals 0. A non-zero number to the power 0 is 1, not 0. So $9^{0} = 1$ .
Believing a rule after trying it on one pair of numbers only. Try a second pair with a different base. A law has to hold for all of them.

Learn and practise “Laws of exponents” in the app →

Grade 8 Exponents

What gets asked

You must be able to

Traps that cost marks

Worked example

Squares, cubes and roots

Rules to remember

Examples

Traps

Exponential form

Rules to remember

Examples

Traps

Laws of exponents

Rules to remember

Examples

Traps

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