Academic Skills Advice
Indices (Powers) & Roots
What do Powers/Indices mean?:
𝑥2
is ‘𝑥 to the power of 2‘ and means 𝑥 × 𝑥
(we usually say ‘𝑥 squared’)
𝑥3
is ‘𝑥 to the power of 3’ and means 𝑥 × 𝑥 × 𝑥
(we usually say ‘𝑥 cubed)
𝑥 4 is ‘𝑥 to the power of 4’ and means 𝑥 × 𝑥 × 𝑥 × 𝑥
etc….
The superscript numbers (2, 3 & 4 above) are known as indices or powers. When the
power is 2 we say “squared”, when the power is 3 we say “cubed” and for all other powers
we say “to the power of….”
Examples:
32 = 3 × 3 = 9
53 = 5 × 5 × 5 = 125
34 = 3 × 3 × 3 × 3 = 81
25 = 2 × 2 × 2 × 2 × 2 = 32
21 = 2
(3 squared)
(5 cubed)
(3 to the power of 4)
(2 to the power of 5)
(2 to the power of 1)
What do roots mean?:
√9 means “which
number multiplied by
itself gives me 9”?
Roots are the opposite of powers.

√𝟗 (square root of 9) has 2 answers √𝟗 = 𝟑 𝑎𝑛𝑑 √𝟗 = −𝟑
We know that 3 × 3 = 32 = 9, so reversing it gives: √9 = 3
Also notice: −3 × −3 = (−3)2 = 9 so also √9 = −3
This is often written as √𝟗 = ±𝟑
(meaning +3 or -3)

𝟑
√𝟐𝟕 (cubed root of 27) = 3
𝟑
We know: 3 x 3 x 3 = 33 = 27, therefore √𝟐𝟕 = 3
Also notice: −3 × −3 × −3 = (−3)3 = −27 so there is only 1 cubed root.

𝟒
√𝟏𝟔 (fourth root of 16) = 2 or -2
4
We know that 2 × 2 × 2 × 2 = 24 = 16, so reversing gives: √16 = 2
4
Also notice: −2 × − 2 × − 2 × − 2 = (−2)4 = 16 so also √16 = -2
𝟒
∴ √𝟏𝟔 = ±𝟐
Remember:
Roots are opposite to powers, therefore a power and it’s root undo each other.
5
e.g. 25 = 32, therefore √32 = 2 (we are back to where we started)
2
(√16) = (4)2 = 16
√(32 ) = √9 = 3
The general formula is:
© H Jackson 2010 / 12 / ACADEMIC SKILLS
𝑛
√𝑥 𝑛 = 𝑥
Notice that if we square then
square root (or vice versa) we
are back to where we started.
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Roots and powers of numbers can be worked out using a calculator, but we need some
rules to help us when we have algebra involved.
The Rules of Indices:
The following are the rules that you need to learn and practice:
Rule
𝒂𝒑 × 𝒂𝒒 = 𝒂𝒑+𝒒
𝒂𝒑
𝒂𝒒
= 𝒂𝒑−𝒒
(𝒂𝒑 )𝒒 = 𝒂𝒑×𝒒
𝒂−𝒑 =
𝒒
𝒑
𝟏
𝒂 = ( √𝒂)
Example
If you multiply 2 numbers with the
same base you add the powers.
35 × 37 = 312
If you divide 2 numbers with the
same base you subtract the powers.
79
= 74
75
If you have a power inside and a
power outside of a bracket you
multiply the powers.
(45 )2 = 410
A negative power means “one over”
so everything is sent to the bottom of
a fraction.
𝒂𝒑
𝒑
What does it mean?
𝒒
𝒂𝟎 = 𝟏
𝒂𝟏 = 𝒂
𝟏𝒂 = 𝟏
9−4 =
1
94
A fractional power means a root. The
bottom of the fraction tells you which
root to take and the top tells you
which power.
164 = ( √16) = 23
Anything to the power of zero = 1
3720 = 1
Any number to the power of 1 stays
the same.
541 = 54
1 to the power of anything = 1
18 = 1
3
4
3
This may seem like a lot to learn but as you practice them they will become easier to
remember.
(See the following pages for some examples.)
© H Jackson 2010 / 12 / ACADEMIC SKILLS
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Examples:

Simplify 𝒙𝟑 × 𝒙𝟐 × 𝒙𝟒
Why do we add the power?
Notice that the X sign is a continuation of what the powers
mean:
𝒙𝟑 × 𝒙𝟐 × 𝒙𝟒
means
𝒙 × 𝒙 × 𝒙 × 𝒙 × 𝒙 × 𝒙 × 𝒙 × 𝒙 × 𝒙 = 𝒙𝟗
The bases (𝑥) are all the same and so we just add the powers:
𝑥 3 × 𝑥 2 × 𝑥 4 = 𝑥 3+2+4

= 𝒙𝟗
Simplify 𝒙𝟓 × 𝒚𝟐 × 𝒙𝟑
Be careful not to combine different bases. We can only add the powers with a base of
𝑥. The base of 𝑦 is different so it stays separate.
𝑥 5 × 𝑦 2 × 𝑥 3 = 𝑥 5+3 × 𝑦 2

= 𝑥8𝑦2
Simplify 𝟑𝒙𝟐 × 𝟒𝒙𝟑
We have numbers as well so just multiply the numbers and add the powers.
3𝑥 2 × 4𝑥 3 = 3 × 4 × 𝑥 2+3
= 12𝑥 5
Why do we subtract the power?
The division could be written as:
25×𝑦×𝑦×𝑦×𝑦×𝑦×𝑦×𝑦
5×𝑦×𝑦×𝑦×𝑦

Notice that the 4 𝑦’s on the bottom will cancel
with 4 from the top leaving 3 𝑦’s on the top.
Simplify 𝟐𝟓𝒚𝟕 ÷ 𝟓𝒚𝟒
Divide the numbers and subtract the powers.
25𝑦 7 ÷ 5𝑦 4 = (25 ÷ 5) (𝑦 7−4 )

= 5𝑦 3
Simplify 𝒙𝟓 ÷ 𝒙−𝟐
Subtract the powers but be careful with the signs.
𝑥 5 ÷ 𝑥 −2 = 𝑥 5−(−2) = 𝑥 5+2

= 𝑥7
Simplify (𝟐𝒙𝟑 )𝟐
Multiply the powers (and remember that 2 is also to the power of 2).
(2𝑥 3 )2 = 22 (𝑥 3 )2
© H Jackson 2010 / 12 / ACADEMIC SKILLS
= 4𝑥 6
3

Simplify 𝒙−𝟓
Remember the rule that the (–) sign in front of a power sends everything to the bottom.
𝟏
𝑥 −5

= 𝒙𝟓
Simplify 𝟑𝒙−𝟓
This is the same as: 3 × 𝑥 −5 .
1
3 × 𝑥 −5 = 3 × 𝑥 5

𝟑
= 𝒙𝟓
𝟏
Simplify 𝒙−𝟐
Now the (–) sign in front of the power is already on the bottom of the fraction so this
time it sends everything to the top.
1
= 𝒙𝟐
𝑥 −2

Write √𝒚 as a power
Remember that roots are fractional powers. This is the square root and so the
denominator of the fraction will be 2.
𝟏
√𝑦

= 𝒚𝟐
𝟒
Write √𝒙 as a power
This is the fourth root and so the denominator of the fraction will be 4.
𝟏
4
√𝑥

= 𝒙𝟒
Write ( 𝟑√𝒚)
𝟐
as a power
This is the third root and so the denominator of the fraction will be 4. It is then
squared and so the numerator will be 2
( 3√𝑦)

2
𝟐
= 𝒚𝟑
Simplify √𝒚 × 𝒚𝟑 × √𝒚
1
1
= 𝑦2 × 𝑦3 × 𝑦2
1
1
= 𝑦 2+3+2
© H Jackson 2010 / 12 / ACADEMIC SKILLS
= 𝑦4
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