exponential functions
exponential functions
Exponent Laws
MCR3U: Functions
In previous courses, we have explored the three
“fundamental” exponent laws:
• the Product Rule,
• the Quotient Rule, and
Exponent Laws
• the Power of a Power Rule.
Integral Exponents
You may have also been introduced to negative or zero
exponents.
J. Garvin
In this unit, we will explore some additional exponent laws
related to rational exponents.
Before we do that, however, we must review the five rules
listed above.
J. Garvin — Exponent Laws
Slide 2/18
Slide 1/18
exponential functions
exponential functions
Multiplying Powers with Like Bases
Dividing Powers with Like Bases
Product Rule of Exponents
Quotient Rule of Exponents
When two powers with the same base are multiplied, add the
exponents.
x a · x b = x a+b
When two powers with the same base are divided, subtract
the exponents.
xa
= x a−b
xb
Consider the powers x a and x b .
x a · x b = x| · x {z
· . . . · x} · x| · x {z
· . . . · x}
a times
b times
= x a+b
Consider the powers x a and x b .
a times
z
}|
{
xa
x · x · x · ... · x
=
b
x| · x {z
· . . . · x}
x
b times
= x a−b
So far, a > b, but this will change shortly.
J. Garvin — Exponent Laws
Slide 4/18
J. Garvin — Exponent Laws
Slide 3/18
exponential functions
exponential functions
Multiplying/Dividing Powers
Raising Powers to an Exponent
Example
Power of a Power Rule
Simplify a5 · a9 .
When one power is raised to another exponent, multiply the
exponents.
(x a )b = x ab
a5 · a9 = a5+9
= a14
Consider the powers x a and x b .
(x a )b = x| a · x a {z
· . . . · x }a
Example
Simplify
b8
b6
b
.
= x ab
b8
= b 8−6
b6
= b2
J. Garvin — Exponent Laws
Slide 5/18
J. Garvin — Exponent Laws
Slide 6/18
times
exponential functions
exponential functions
Raising Powers to an Exponent
Raising Powers to an Exponent
Example
Example
8
Simplify p 2 q 3 .
Simplify
p2q3
8
a2 b 4
c3
5
.
= p 2·8 q 3·8
= p 16 q 24
J. Garvin — Exponent Laws
Slide 7/18
a2 b 4
c3
5
a2·5 b 4·5
c 3·5
10
a b 20
=
c 15
=
J. Garvin — Exponent Laws
Slide 8/18
exponential functions
exponential functions
Zero Exponents
Zero Exponents
Zero Exponent Rule
What about 00 ?
Any non-zero base with a zero exponent has a value of 1.
Using the quotient rule, 00 =
x 0 = 1, x 6= 0
This can be shown using the quotient rule described earlier.
xn
Consider the expression n . Using the quotient rule,
x
xn
= x n−n = x 0 .
xn
xn
At the same time, n indicates that x n is divided by itself,
x
which is clearly 1.
J. Garvin — Exponent Laws
Slide 9/18
0n
.
0n
Since this expression involves division by zero, 00 is not
defined.
J. Garvin — Exponent Laws
Slide 10/18
exponential functions
exponential functions
Zero Exponents
Negative Exponents
Example
Negative Exponent Rule
Simplify (3 − 7)0 .
A power with a negative exponent is equivalent to the
reciprocal of that power with a positive exponent.
Since the expression inside of the brackets is not zero,
(3 − 7)0 = 1.
Example
Simplify (2x − 1)0 .
x −n =
We can demonstrate this to be true by using zero exponents.
x −n = x 0−n
If x = 21 , then the expression inside of the brackets will be
zero, resulting in 00 . For all other values of x, this problem
does not arise.
x0
xn
1
= n
x
=
We can say (2x − 1)0 = 1, x 6= 12 . We have seen restrictions
on the variable before, when we covered rational expressions.
J. Garvin — Exponent Laws
Slide 11/18
1
xn
J. Garvin — Exponent Laws
Slide 12/18
exponential functions
exponential functions
Negative Exponents
Combining Exponent Laws
Example
The previous exponent laws can be combined to simplify
expressions to a single power.
Express
a−5
using a positive exponent.
a−5 =
Most of the time, it is preferable to express answers using
positive exponents.
Example
Express
Whenever possible, look for shortcuts like reducing or
combining powers first.
1
a5
ab
c
−3
using positive exponents.
ab
c
=
=
1
ab 3
c
c3
a3 b 3
J. Garvin — Exponent Laws
Slide 14/18
J. Garvin — Exponent Laws
Slide 13/18
exponential functions
exponential functions
Combining Exponent Laws
Combining Exponent Laws
Example
Example
2
Simplify x 3 · x 4 .
Simplify
x3 · x4
2
= x 3+4
2
= x7
2
n2 ·n5
n4
2 8 = x 7·2
= x 14
n
n6
n2 · n5
n4
.
2 n8
n6
2 8 n7
n
n4
n6
2 2 = n3
n
=
= n6 · n2
= n8
J. Garvin — Exponent Laws
Slide 16/18
J. Garvin — Exponent Laws
Slide 15/18
exponential functions
Combining Exponent Laws
Example
Simplify
exponential functions
Questions?
4
p 3 q −2
using positive exponents only.
p 2 q −3
4
p 3 q −2
p 12 q −8
= 2 −3
2
−3
p q
p q
= p 10 q −5
=
J. Garvin — Exponent Laws
Slide 17/18
p 10
q5
J. Garvin — Exponent Laws
Slide 18/18