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Working with Exponents
MPM1D: Principles of Mathematics
Recap
Simplify, then evaluate, 24 × 26 .
Perform exponentiation before multiplication.
24 × 26 = 16 × 64
Exponent Laws (Numerical Bases)
= 1 024
J. Garvin
Note that 1 024 = 210 and that 24+6 = 210 .
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Exponent Laws
Exponent Laws
An expression involving some value (or base) raised to some
exponent is called a power.
In the earlier example, 24 × 26 , both powers have the same
base.
23
base → |{z}
power
Their product is also a power of 2, and its exponent is the
sum of the two given exponents.
← exponent
This is easy to see when each power is written in expanded
form.
24
The resulting value is often referred to as a “power of” the
base. For example, 23 = 8, so we might say that 8 is a power
of 2.
Some people also use the term “power” to refer to the
exponent, e.g. “2 to the power of 3”, but it is probably more
accurate to use the phrase “2 to the exponent 3” instead.
26
z
}|
{ z
}|
{
2 × 2 = |2 × 2 × 2 × 2 × 2 {z
× 2 × 2 × 2 × 2 × 2}
4
6
10 times
= 210
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Exponent Laws
Exponent Laws
This property is true for the product of any two powers with
the same base.
Example
Product Rule for Exponents
For any two powers with the same base, their product is a
power with the same base and an exponent equal to the sum
of the given exponents.
Simplify, then evaluate, 32 × 33 .
Since there is a common base of 3, add the exponents.
32 × 33 = 32+3
= 35
This allows us to simplify expressions involving multiple
powers with the same base.
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= 243
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Exponent Laws
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Exponent Laws
Example
A similar rule exists when powers with like bases are divided.
Simplify, then evaluate,
28
×
210 .
Consider the expression
Since there is a common base of 2, add the exponents.
8
10
2 ×2
37
. In expanded form, this is
35
3×3×3×3×3×3×3
3×3×3×3×3
8+10
=2
= 218
We can cancel each 3 in the denominator with a matching 3
in the numerator.
= 262 144
3 × 3 × 3 × 3 × 3 × 3 × 3
3 × 3 × 3 × 3 × 3
This leaves us with 3 × 3, or 32 .
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Exponent Laws
Exponent Laws
37
Note that 5 = 32 , and that 37−5 = 32 .
3
Example
Quotient Rule for Exponents
Simplify, then evaluate,
For any two powers with the same base, their quotient is a
power with the same base and an exponent equal to the
difference of the given exponents.
Since there is a common base of 10, subtract the exponents.
109
.
106
109
= 109−6
106
= 103
It is possible to obtain negative exponents using this property.
We will talk about these in more detail in later courses.
= 1 000
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Exponent Laws
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Exponent Laws
Example
Simplify, then evaluate, 154 ÷ 153 .
Since there is a common base of 15, subtract the exponents.
3
What about the expression 52 ?
3
Expanding this, we get 52 = 52 × 52 × 52 .
Using the product rule, we can add the exponents, since
there is a common base of 5.
154 ÷ 153 = 154−3
52 × 52 × 52 = 52+2+2
= 151
= 56
= 15
Note that 52
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3
= 56 and that 52×3 = 56 .
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Exponent Laws
Exponent Laws
This is a third exponent law, applying to a “power of a
power”.
Example
Power of a Power Rule for Exponents
For any power raised to an exponent, its value is a power
with the same base and an exponent equal to the product of
the given exponents.
3
Simplify, then evaluate, 25 .
Multiply the exponents.
25
There are other exponent laws in addition to these three, but
they will be covered in later courses.
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3
= 25×3
= 215
= 32 768
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Exponent Laws
Exponent Laws
Sometimes, expressions involve multiple exponent laws.
Example
They can usually be applied in any order, but try to keep
things simple – make values smaller, rather than larger, and
try to cancel things as early as possible.
2
Simplify 26 × 23 .
Add the exponents inside of the brackets first, then multiply.
26 × 23
2
= 26+3
2
= 29
2
= 29×2
= 218
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Questions?
Example
Simplify
513
511
4
.
Subtract the exponents inside of the brackets first, then
multiply.
513
511
4
= 513−11
= 52
4
4
= 52×4
= 58
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