2. The World’s Population is growing at an Exponential Rate.
Most of the growth is in developing countries where many people
are having many children. How will they all be accommodated?
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3. If we have One Person and they have 4 children, and then
each of these children have 4 children, and so on, we get the
following “Power of Power” Exponential Population Growth.
Generation 0 1 2 3 4
Children 1 4 16 64 216
Powers Values (20)2
(21)2
(22)2
(23)2
(24)2
Rule: (2Generation)2
Children 1 4 16 64 216
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4. The big number “2” is called the “base”
and is what we multiply together
The little number “4” is called the
“Index” or “Exponent” and tells us
how many times to multiply out
the big number “2”
24
= 2 x 2 x 2 x 2
Multiply four of the Base Number
5. Now apply the “Add Rule” for Multiplication
The Power of “4” outside the
brackets, tells us to multiply
out what is contained in the
brackets four times.
(23
) 4
= 23
x 23
x 23
x 23
Multiply four lots of what is in the brackets
= 23 + 3 + 3 + 3
= 212
6. Now apply the “Add Rule” for Multiplication
The same approach can be used
for expanding an Algebra variable
letter, with a Power of “4” outside
the brackets.
(n2
) 4
= n2
x n2
x n2
x n2
Multiply four lots of what is in the brackets
= n2 + 2 + 2 + 2
= n8
7. For the brackets item (am
)
n
If we have a Base “a”, raised to a Power, and
then we raise this to another Power, we can
skip expanding out the powers and use the
fast track rule and MULTIPLY the Exponents
(am
)n
= am x n
or amn
This rule works for both letters and numbers
8. 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.
9. 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
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10. Let’s consider (22
)3
x (24
)5
We apply the Power Rule to both items:
(22
)3
x (24
)5
= 22 x 3
x 24 x 5
= 26
x 220
= 220 + 6
= 226
We now finish our task
by using the ADD RULE
for Multiplying Power
terms which have the
exact same Base.
11. Simplify the expression (m3
)2
x (m2
)5
We apply the Power Rule to both items:
(m3
)2
x (m2
)5
= m3 x 2
x m2 x 5
= m6
x m10
= m10 + 6
= m16
We now finish our task
by using the ADD RULE
for Multiplying Power
terms which have the
exact same Base.
12. Simplify the expression (p3
)2
x (q2
)5
We apply the Power Rule to both items:
(p3
)2
x (q2
)5
= p3 x 2
x q2 x 5
= p6
x q10
= p6
q10
We CANNOT use the
ADD RULE because the
Power terms do NOT
have the same Base.
13. Simplify using Exponent Rules (32
)4
(33
)2
We apply the Power Rule to both items:
(32
)4
(33
)2
= 32 x 4
33 x 2
= 38
36
= 38 - 6
= 32
We now finish our task
by using SUBTRACT
RULE for Dividing
terms which have the
exact same Base.
14. Simplify using Exponent Rules (h4
)3
(h2
)2
We apply the Power Rule to both items:
(h4
)3
(h2
)2
= h4 x 3
h2 x 2
= h12
h4
= h12 - 4
= h8
We now finish our task
by using SUBTRACT
RULE for Dividing
terms which have the
exact same Base.
15. Simplify the expression (k3
)2
(w2
)5
We apply the Power Rule to both items:
(k3
)2
(w2
)5
= k3 x 2
w2 x 5
= k6
w10
= k6
w10
We CANNOT use the
SUBTRACT RULE
because the Power
terms do NOT both
have the same Base.
