Exponents (Grade 10) — Mathematics Grades 8–11

Apply the exponent laws to simplify expressions and solve exponential equations.

Gr 10 Term 1 - ExponentsTerm 1 - 2 weeks (CAPS)

Practise Exponents in the app →

What gets asked

Simplify an expression using two or more exponent laws.
Rewrite an expression so that every exponent is positive.
Solve an exponential equation by writing both sides with the same base.
Convert between surd form and a rational (fraction) exponent.

You must be able to

Apply the product, quotient and power laws to powers with the same base.
Move a factor across the fraction bar to make its exponent positive.
Rewrite a number like 9 or 27 as a power of 3 before comparing exponents.
Write a cube root as a power of one third, and change it back again.

Traps that cost marks

Treating the whole term as raised to zero, so $3 x^{0}$ is read as $1$ . The exponent only touches what it is attached to. $3 x^{0} = 3 \times 1 = 3$ .
Adding the exponents of $9^{x}$ and $3^{x}$ directly, as if the bases already match. Rewrite $9^{x}$ as $3^{2 x}$ first. Only then can the exponents combine.
Setting two exponents equal before the bases have been rewritten the same. Rewrite every term with one base first. Only then may you equate the exponents.

Worked example

Simplify:

(2 x^{3})^{2} \times x^{- 4}

Square the bracket first: $(2 x^{3})^{2} = 4 x^{6}$ .
Multiply the powers of $x$ : $x^{6} \times x^{- 4} = x^{2}$ .
Answer: $4 x^{2}$ .

Laws of exponents

The rules for multiplying, dividing and raising powers, and what a zero or a minus up there means.

base: The number or letter being multiplied by itself. 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 its exponent written together, like $2^{5}$ or $x^{3}$ .
coefficient: The number multiplying a power. In $3 x^{2}$ the coefficient is $3$ .
reciprocal: One divided by the number. The reciprocal of $8$ is $\frac{1}{8}$ .

A power has two parts. In $2^{5}$ the base is $2$ and the exponent is $5$ . It tells you to multiply five $2$ s together, so $2^{5} = 32$ . The base does the multiplying and the exponent does the counting.

Multiplying powers with the same base adds the exponents. Dividing them subtracts. The base itself never changes. So $x^{4} \times x^{3} = x^{7}$ . Different bases cannot be combined this way at all.

A power raised to another power multiplies the exponents. A bracket raised to a power reaches every factor inside, coefficient included. So $(3 x^{2})^{3} = 27 x^{6}$ , not $3 x^{6}$ .

Any base except $0$ , raised to the exponent $0$ , gives $1$ . But the exponent only touches what it is attached to. In $3 x^{0}$ the zero sits on the $x$ alone, so $3 x^{0} = 3$ .

A negative exponent never makes a value negative. It means the reciprocal, so $2^{- 3} = \frac{1}{8}$ . To clear one, move the whole power across the fraction bar. Anything without a negative exponent stays where it is.

Rules to remember

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

Examples

Simplify:

\frac{x ^{7} \times x ^{2}}{x ^{3}}

Worked answer

Do the top first: $x^{7} \times x^{2} = x^{9}$ .
Now divide: $x^{9} \div x^{3} = x^{6}$ .
The base stayed $x$ ; only the exponents moved.

Answer: $x^{6}$

Adding and then subtracting exponents kept one base from start to finish.

Simplify:

(2 a^{3})^{4} \times a^{0}

Worked answer

The outside exponent reaches both parts: $(2 a^{3})^{4} = 2^{4} a^{12}$ .
$2^{4} = 16$ , so that is $16 a^{12}$ .
$a^{0} = 1$ , so multiplying by it changes nothing.
The answer is $16 a^{12}$ .

Answer: $16 a^{12}$

The coefficient was raised to the power as well, and the zero exponent gave $1$ .

Write with positive exponents only:

\frac{3 x ^{- 2}}{y ^{- 1}}

Worked answer

$x^{- 2}$ is on top, so it moves down and becomes $x^{2}$ .
$y^{- 1}$ is on the bottom, so it moves up and becomes $y$ .
The coefficient $3$ has no negative exponent, so it stays.
That gives $\frac{3 x ^{- 2}}{y ^{- 1}} = \frac{3 y}{x ^{2}}$ .

Answer: $\frac{3 y}{x ^{2}}$

Only the parts carrying a negative exponent crossed the fraction bar.

Traps

Multiplying the bases too, so $2^{3} \times 2^{4}$ becomes $4^{7}$ . The base stays put and only the exponents add. So $2^{3} \times 2^{4} = 2^{7}$ .
Adding the exponents for a power of a power, so $(x^{3})^{4}$ becomes $x^{7}$ . A power of a power multiplies them. So $(x^{3})^{4} = x^{12}$ .
Treating the whole term as raised to zero, so $3 x^{0}$ is read as $1$ . The exponent only touches what it is attached to. $3 x^{0} = 3 \times 1 = 3$ .
Reading $2^{- 3}$ as a negative number. A negative exponent means the reciprocal. So $2^{- 3} = \frac{1}{8}$ , which is positive.

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

Simplify and factorise

Getting every part of an expression onto one base, and spotting the factors hiding in it.

prime: A number with exactly two factors, $1$ and itself. $2$ , $3$ and $5$ are primes.
common base: One base that every power in the expression can be rewritten with. Both $9$ and $27$ take base $3$ .
lowest power: The smallest power of the base that appears. It is the one you take out as a common factor.
difference of two squares: One square subtracted from another, of the form $a^{2} - b^{2}$ .

Work in order. Deal with a bracket and its outside exponent first. Then multiply or divide the powers that share a base. Tidy the plain numbers last. Doing it out of order can change the answer.

Numbers that look different often are not. $9$ is $3^{2}$ and $27$ is $3^{3}$ , so both take the common base $3$ . Pick a prime as that base whenever you can.

Rewriting a base changes the exponent as well. $9^{x}$ becomes $3^{2 x}$ , never $3^{x}$ . The $2$ that comes from $9 = 3^{2}$ multiplies the exponent that was already there.

To take out a common factor, take out the lowest power. In $2^{x + 3} - 2^{x}$ that is $2^{x}$ . Take out a bigger one and you are left with negative exponents inside the bracket.

An even exponent means a square is hiding. Since $4^{x} = (2^{x})^{2}$ , the expression $4^{x} - 9$ is a difference of two squares. To find what is being squared, halve the exponent and leave the base alone.

Rules to remember

$a^{m + n} = a^{m} \times a^{n}$
$a^{2} - b^{2} = (a + b) (a - b)$

Examples

Simplify:

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

Worked answer

Do the bracket first: $(2 x^{2})^{3} = 8 x^{6}$ .
Divide the numbers: $8 \div 4 = 2$ .
Divide the powers: $x^{6} \div x^{4} = x^{2}$ .
So the answer is $2 x^{2}$ .

Answer: $2 x^{2}$

Expanding the bracket first is what left something worth dividing.

Simplify:

\frac{2 ^{x + 3} - 2 ^{x}}{2 ^{x}}

Worked answer

Both top terms carry a power of $2$ , and the lowest power is $2^{x}$ .
$2^{x + 3} = 2^{x} \times 2^{3}$
So the top is $2^{x} (2^{3} - 1) = 2^{x} \times 7$ .
Divide by $2^{x}$ , and $7$ is all that is left.

Answer: $7$

Taking out the lowest power left plain numbers inside the bracket.

Simplify:

\frac{9 ^{x} - 4}{3 ^{x} + 2}

Worked answer

Write the first term with base $3$ : $9^{x} = (3^{x})^{2}$ .
And $4 = 2^{2}$ .
So the top is a difference of two squares.
It factorises into $(3^{x} - 2) (3^{x} + 2)$ .
Cancel $(3^{x} + 2)$ , which leaves $3^{x} - 2$ .

Answer: $3^{x} - 2$

Halving the exponent, and not the base, is what showed up the square.

Traps

Adding the exponents of $9^{x}$ and $3^{x}$ directly, as if the bases already match. Rewrite $9^{x}$ as $3^{2 x}$ first. Only then can the exponents combine.
Taking out the highest power, so negative exponents appear inside the bracket. Take out the lowest power instead. In $2^{x + 3} - 2^{x}$ that is $2^{x}$ .
Halving the base rather than the exponent, so $4^{x}$ is read as $2^{x}$ . The base stays and the exponent halves. So $4^{x} = (2^{x})^{2}$ .
Adding exponents across two different letters, so $x^{2} y^{3}$ becomes $x y^{5}$ . Only powers with the same base combine. $x^{2} y^{3}$ is already as simple as it gets.

Learn and practise “Simplify and factorise” in the app →

Exponential equations

Solving for a letter that is sitting up in the exponent instead of down on the line.

exponential equation: An equation with the unknown up in the exponent, like $2^{x} = 8$ .
equate the exponents: To write the two exponents as equal to each other, once both sides carry the same base.
solution: The value of the unknown that makes the equation true.

In an exponential equation the unknown sits in the exponent. You cannot get at it by dividing, because the base is not a coefficient. Instead, make both sides into powers of one base.

Once the bases match, the exponents must match too. A power such as $2^{x}$ takes each value only once, so nothing else can give the same answer. That is why $2^{x} = 2^{5}$ forces the solution $x = 5$ .

If the bases differ, rewrite them first. $8 = 2^{3}$ , so $8^{x}$ becomes $2^{3 x}$ . Do not forget to multiply the exponent that was already there.

Fractions and negatives are handled the same way. $\frac{1}{9} = 3^{- 2}$ , so a fraction just needs a negative exponent. Write both sides over one base, then equate the exponents.

Check your answer by putting it back in. Substituting the solution into the original equation should give a true statement, and it takes ten seconds.

Rules to remember

If $a^{m} = a^{n}$ , and $a$ is positive and not $1$ , then $m = n$
$a^{- n} = \frac{1}{a ^{n}}$

Examples

Solve for

x

2^{x} = 32

Worked answer

Write $32$ as a power of $2$ : $32 = 2^{5}$ .
The equation is now $2^{x} = 2^{5}$ .
The bases match, so equate the exponents.
That gives $x = 5$ .

Answer: $5$

Both sides had to carry base $2$ before the exponents could be compared.

Solve for

x

9^{x} = 27

Worked answer

Both numbers are powers of $3$ : $9 = 3^{2}$ and $27 = 3^{3}$ .
So $9^{x} = 3^{2 x}$ .
The equation becomes $3^{2 x} = 3^{3}$ .
Equate the exponents: $2 x = 3$ .
So $x = \frac{3}{2}$ .

Answer: $\frac{3}{2}$

Rewriting $9$ as $3^{2}$ doubled the exponent, which is the easy thing to miss.

Solve for

x

3^{- x} = \frac{1}{81}

Worked answer

$81 = 3^{4}$ , so $\frac{1}{81} = 3^{- 4}$ .
The equation is now $3^{- x} = 3^{- 4}$ .
Equate the exponents: $- x = - 4$ .
So $x = 4$ .

Answer: $4$

A fraction below $1$ needed a negative exponent before the bases could match.

Traps

Equating the bases instead of the exponents, so $2^{x} = 32$ gives $x = 2$ . Rewrite first, then compare exponents. From $2^{x} = 2^{5}$ you read off $x = 5$ .
Setting two exponents equal before the bases have been rewritten the same. Rewrite every term with one base first. Only then may you equate the exponents.
Rewriting $9^{x}$ as $3^{x}$ and losing the doubling. Since $9 = 3^{2}$ , the exponent doubles, giving $9^{x} = 3^{2 x}$ .
Writing $\frac{1}{81}$ as $3^{4}$ and dropping the minus sign. A number below $1$ needs a negative exponent, so $\frac{1}{81} = 3^{- 4}$ .

Learn and practise “Exponential equations” in the app →

Surds and rational exponents

A root sign and a fraction exponent are two ways of writing the same thing.

surd: A root whose value is irrational, like $5$ or $37$ . $4$ is not a surd — it works out to $2$ — so a root sign alone does not make one.
index: The small number sitting in the root sign, saying which root it is. In $38$ the index is $3$ .
square root: The number that multiplies by itself to give the number under the sign. $25 = 5$ because $5 \times 5 = 25$ .
rational exponent: An exponent that is a fraction, such as $\frac{1}{2}$ or $\frac{2}{3}$ .

Every surd can be written as a power with a rational exponent. A square root has index $2$ , so $x = x^{\frac{1}{2}}$ . A cube root has index $3$ , so $3 x = x^{\frac{1}{3}}$ . The index always lands underneath.

If there is also a power inside the root, it goes on top of the fraction. So $3 x^{2} = x^{\frac{2}{3}}$ . The top is the power and the bottom is the root. Reading that fraction the wrong way round is the usual mistake.

The root reaches everything under the sign, numbers included. In $9 x$ the $9$ is under the sign too, so $9 x = 3 x$ . Do not take the root of the letter and leave the number behind.

For a positive base, all the exponent laws still work when the exponents are fractions. The positive base matters: fraction powers of a negative base are not defined here, and the laws can fail for them. Multiplying powers of the same base still adds the exponents, and dividing still subtracts them. The fractions just need a common denominator first.

With numbers, take the root before the power. For $1 6^{\frac{3}{4}}$ , take the fourth root of $16$ to get $2$ , then cube it. Working that way round keeps the numbers small enough to do in your head.

Rules to remember

$a^{\frac{1}{n}} = n a$
$a^{\frac{m}{n}} = n a^{m}$

Examples

Write

3 x^{2}

as a power of

x

Worked answer

The index is $3$ , so the bottom of the exponent is $3$ .
The power inside the root is $2$ , so the top is $2$ .
That gives $3 x^{2} = x^{\frac{2}{3}}$ .

Answer: $x^{\frac{2}{3}}$

The index goes under the line and the inside power goes over it.

Work out:

1 6^{\frac{3}{4}}

Worked answer

The bottom $4$ says take the fourth root: $416 = 2$ .
The top $3$ says cube that answer: $2^{3} = 8$ .
So $1 6^{\frac{3}{4}} = 8$ .

Answer: $8$

Rooting first kept the numbers small, since $1 6^{3}$ would have been huge.

Simplify:

x^{\frac{1}{2}} \times x^{\frac{1}{3}}

Worked answer

The base is the same, so add the exponents.
$\frac{1}{2} + \frac{1}{3} = \frac{5}{6}$
So the answer is $x^{\frac{5}{6}}$ .
In surd form that is $6 x^{5}$ .

Answer: $x^{\frac{5}{6}}$

The two fractions needed a common denominator before they could be added.

Traps

Putting the index on top, so $3 x$ is written as $x^{3}$ . The index belongs underneath, so $3 x = x^{\frac{1}{3}}$ .
Rooting the letter only, so $9 x$ is written as $9 x$ . The root reaches the whole inside, so $9 x = 3 x$ .
Adding $\frac{1}{2}$ and $\frac{1}{3}$ to get $\frac{2}{5}$ . Use a common denominator: $\frac{1}{2} + \frac{1}{3} = \frac{5}{6}$ .
Dividing the fraction exponents when dividing two powers. Subtract them instead: $x^{\frac{3}{4}} \div x^{\frac{1}{4}} = x^{\frac{1}{2}}$ .

Learn and practise “Surds and rational exponents” in the app →

Grade 10 Exponents

What gets asked

You must be able to

Traps that cost marks

Worked example

Laws of exponents

Rules to remember

Examples

Traps

Simplify and factorise

Rules to remember

Examples

Traps

Exponential equations

Rules to remember

Examples

Traps

Surds and rational exponents

Rules to remember

Examples

Traps

About this material