This document provides rules and examples for operations involving exponents. It explains that when bases are the same, exponents can be added or subtracted. It also discusses the power of a power rule where the outer exponent is multiplied by the inner one. There are warnings that these rules only apply when bases are the same or there is a single base inside brackets. Negative exponents are explained as reciprocals with the base moving above or below the fraction line.

2. Let’s consider am
x an
If we have any base that is the same “a”,
then we can skip expanding out the powers
of the exponents “m’ and “n”, and use the
fast track rule which is to Add the Exponents
am
x an
= am + n
This rule works for both letters and numbers

3. WARNING: We cannot combine the Power
Terms and add their exponents if the big
number or letter Bases are different !
23
x m4
= 2m 3+4
= 2m7
Different Bases It is wrong to combine them like this
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4. Let’s consider am
an
or am
an
If we have any base that is the same “a”,
then we can skip expanding and cancelling
the exponents “m’ and “n”, and use the fast
track rule and SUBTRACT the Exponents
am
an
= am - n
This rule works for both letters and numbers

5. The Power of Power
Rule involves Multiplying
the two Index Powers.
(23
)
4
= 23 x 4
= 212
(n2
)
4
= n2 x 4
= n8
This rule only works if there is a single Positive Base inside the brackets.

6. WARNING: The Power of Power Rule only
works if there is one single positive Base
(eg. a number or letter) inside the brackets!
(2n3
)4
= 2n3x4
= 2n12
Two Bases It is wrong to expand them like this
Images from Clker.com

7. Eg. The 2 and the a BOTH need to raised to 2.
The Power outside the brackets needs to be applied to all Bases
Inside the brackets. (Like the Distributive Law, but for Exponents).
(2a)
2
= 22
x a2
= 4 x a2
= 4a2

8. We can use the Expanding Products Rule BACKWARDS to
simplify expressions of Different Bases but Same Index Number:
2 4
x 5 4
= (2 x 5)4
= 104
m 3
x t 3
= (m x t)3
= (mt)3

9. Eg. The a and the b BOTH need to be Powered.
The Expanding Quotients Rule involves
applying the Power Outside of the
brackets, onto every item that is inside
the brackets.

10. Eg. Different Top and Bottom, but same Powers.
The Expanding Quotients Rule can also
be used in reverse, to make a fraction
with IDENTICAL POWERS into a
single bracketed exponent Fraction.

11. 999990
1=
Any Value to the Power of Zero Equals 1 : a0
= 1
m0
1=
3k0
3 x k0
3 x 1 3= = =
50
b2
50
x b2
1 x b2
b2
= = =
4a2
b0
4 x a2
x b0
4a2
= =

12. Note “a” cannot be zero, because 1/0 is not possible .
A Negative exponent means we have to
re-write our Power term as a 1/ Fraction.
Negative Exponents are Positive Fractions.

13. Due to the way flipped over fractions called “Reciprocals” work:
An item in the TOP with a Negative Index Power moves to the
BOTTOM , where it becomes a POSITIVE Index Power.
An item in the BOTTOM with a Negative Index Power moves to
the TOP, where it becomes a POSITIVE Index Power.
5-2
= 1/52
= 1/25 but 1/5-2
= 52
/1 = 25
2-3
54
5-4
23= 22
22
x 32
3-2 =
4 x 9 = 36=

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