polynomials
polynomials
Exponent Laws
MPM2D: Principles of Mathematics
Consider the expression x 2 · x 3 .
Using the definition of exponentiation, x 2 · x 3 can be
expressed as (x · x)(x · x · x) = x · x · x · x · x = x 5 .
More generally, x a · x b = (x · x · . . . · x) · (x · x · . . . · x)
|
{z
} |
{z
}
Exponent Laws
= x| · x {z
· . . . · x} = x a+b .
a times
b times
a+b times
J. Garvin
Product of Like Powers Law
For any real, non-zero values a, b and x, x a · x b = x a+b .
If the bases are not the same, this rule does not apply. The
expression 24 · 32 cannot be simplified further.
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polynomials
polynomials
Exponent Laws
Exponent Laws
x3
Next, consider the expression 2 .
x
x3
x ·x ·x
Rewriting, 2 can be expressed as
= x.
x
x ·x
(x · x · . . . · x)
|
{z
}
xa
a times
More generally, b =
= x · x {z
· . . . · x} = x a−b .
(x · x · . . . · x) |
x
|
{z
}
a-b times
2
Now, consider x 3 .
2
Rewriting, x 3 becomes (x · x · x) · (x · x · x) = x 6 .
b times
Quotient of Like Powers Law
In general, (x a )b =
(x| · x {z
· . . . · x}) · (x| · x {z
· . . . · x}) · . . . · (x| · x {z
· . . . · x}) = x ab .
a times
a times
a times
|
{z
}
b times
Power of a Power Law
For any real, non-zero values a, b and x, (x a )b = x ab .
xa
For any real, non-zero values a, b and x, b = x a−b .
x
Like the earlier Product Law, the bases must be the same.
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polynomials
polynomials
Exponent Laws
Exponent Laws
Consider (xy )2 next.
Like the power of a product, the power of a quotient can be
similarly defined.
2 x
x
x
x2
For instance,
=
·
= 2.
y
y
y
y
a x
x
x
x
xa
In general,
=
·
· ... ·
= a
y
y
y
y
y
|
{z
}
In its longer form, (xy )2 = (xy )(xy ) = (x · x)(y · y ) = x 2 y 2 .
In general, (xy )a = (xy ) · (xy ) · . . . · (xy ) =
|
{z
}
a times
(x · x · . . . · x) · (y · y · . . . · y ) = x a y a .
{z
} |
{z
}
|
a times
a times
a times
Power of a Product Law
For any real, non-zero values a, x and y ,
(xy )a
=
x ay a.
Power of a Quotient Law
For any real, non-zero values a, x and y ,
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a
x
xa
= a.
y
y
polynomials
Exponent Laws
polynomials
Exponent Laws
What about the expression x 0 ?
Example
Simply the expressions x 4 · x 7 ,
z8
z
, k
6
x 4 · x 7 = x 4+7 = x 11 .
z8
= z 8−6 = z 2 .
z6
5
k 3 = k 3×5 = k 15 .
3 5
, (2p)3 and
x 5
2
.
According to the Quotient Law,
xa
= x a−a = x 0 .
xa
k
= 1, as long as k 6= 0.
k
k
xa
If k = x a , then = a = x 0 = 1.
k
x
At the same time,
Zero Exponent Law
(2p)3 = 23 p 3 = 8p 3 .
x 5 x 5
x5
= 5 = .
2
2
32
For any real, non-zero value of x, x 0 = 1.
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polynomials
Exponent Laws
Exponent Laws
x −2 ,
What does a negative exponent, like
mean?
x
x
1
1
1−3
−2
Since 3 = x
= x , and since 3 = 2 , then x −2 = 2 .
x
x
x
x
In general, x a · x −a = x a+(−a) = x 0 = 1, assuming x 6= 0.
Therefore,
1
x −a = a .
x
xa
·
x −a
polynomials
= 1, which can be rearranged to
Example
Evaluate 1 234 5670 .
Since the base is non-zero, 1 234 5670 = 1.
Example
Express x −4 using positive exponents.
x −4 =
Negative Exponent Law
For any real, non-zero value of x and any real, positive value
1
of a, x −a = a .
x
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1
. Again, x cannot equal zero.
x4
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polynomials
polynomials
Exponent Laws
Exponent Laws
Sometimes it is necessary to combine two or more exponent
laws to simplify an expression.
Example
Simplify (5p −3 q)−2 , using positive exponents.
Example
Simplify
x 5y 3
, using positive exponents.
x 2y 7
(5p −3 q)−2 = 5−2 p (−3)(−2) q −2
1
1
= 2 · p6 · 2
5
q
p6
=
25q 2
x 5y 3
= x 5−2 y 3−7
x 2y 7
= x 3 y −4
=
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x3
y4
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polynomials
polynomials
Scientific Notation
Scientific Notation
Scientific notation is a system, used in many sciences, that
expresses numbers using powers of 10.
Example
For example, the number 352 can be expressed as 3.52 × 102 ,
since 3.52 × 102 = 3.52 × 100 = 352.
It is often used as a shorthand notation for very small or very
large numbers.
For instance, 3 800 000 000 000 (3 trillion, 800 billion) can be
expressed more simply as 3.8 × 1012 .
By convention, scientific notation expresses all numbers with
one digit before the decimal point – that is, 4.3 × 103 rather
than 43 × 102 .
Positive exponents indicate the decimal point has been
shifted left, while negative exponents indicate a right shift.
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Shifting the decimal 7 places to the left, and rounding down,
75 328 143 = 7.53 × 107 .
Example
Express 0.000 031 874 using scientific notation, to two
decimal places.
Shifting the decimal 5 places to the right, and rounding up,
0.000 031 874 = 3.19 × 10−5 .
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polynomials
Questions?
Express 75 328 143 using scientific notation, to two decimal
places.