Exponents Summary
Zero Exponents
a0 = 1
Negative Exponents
a-n = 1 / an
Examples:
A) -5-2 = 1 / -52
b)
5a3b-2 =
5a3(1/b2) = 5a3/b2
What do I do if there is a negative exponent in the denominator (bottom)?
Example:
1 / x-5 = 1 / (1 / x5)
Next, we know that when we divide by a fraction, it is the same thing as multiplying by the reciprocal. So
flip the bottom and multiply it by the top:
1 × x5 = x5
Multiplying Powers with the Same Base
am × an = am+n
When youMULTIPLY the same bases with different exponents, just ADD the exponents
Hey – what if there are coefficients? Then multiply the coefficients.
Example:
3a2 × 4a6 = 12a8
When you are multiplying and there are negative exponents, add or subtract the exponents before you
make any fractions. You may not have to make a fraction. Here are 2 examples with each type:
4z5 × 9z-3
36z2
Do not put z-3 into the denominator. Add or subtract the exponents first:
jNext example:
4z5 × 9z-12 Again, add or subtract the exponents first (and multiply the coefficients)
36z-7
Next, we form the fraction, but the 36 stays in the numerator (top), so we get:
36 / z7
Raising a Power to a Power
(am)n = amn
Example: (a3)2 = a3x2 = a6
What if there is a COEFFICIENT inside the parenthesis?
That coefficient must also be raised to the power of the exponent outside the coefficient:
(ab)n = anbn
(3×2)3 = 3323 = 27 x 8 = 216 This is also equal to:
63 = 216
Big Example:
Simplify: (n1/2)10(4mn-2/3)3
(n1/2)1043m3(n-2/3)3
First, raise each factor of 4mn-2/3 to the 3rd power
n543m3n-2
Multiply the exponents of a power raised to a power
43m3n3
Notice there is an n5 and an n-2. Add or Subtract exponents of
Powers with the same base.
64m3n3
Simplify.
Dividing Exponents
am / an = am-n
To divide powers with the same base, subtract the exponents
What if I have to raise a fraction, also known as a quotient, to a power?
You raise both the numerator and denominator to the power and then simplify.
Rule: (a/b)n = an / bn
Example:
(3/5)3 = 33/53 = 27/125
Big example:
Simplify: (2x6 / y4)-3
____1______
(2x6 / y4)3
=
Since the -3 exponent indicates that we will make this 1 over the whole
Expression, we can actually convert this to the reciprocal:
(y4/2x6)3
(y4)3 / (2x6)3
Raise the numerator and denominator to the 3rd power
y12 / 8x18
Simplify
SCIENTIFIC NOTATION
Scientific Notation (also called Standard Form in Britain) is a special way of writing numbers:
Example: 700
Why is 700 written as 7 × 102 in Scientific Notation ?
700 = 7 × 100
and 100 = 102 (see powers of 10)
so 700 = 7 × 102
Both 700 and 7 × 102 have the same value, just shown in different ways.
Example: 4,900,000,000
1,000,000,000 = 109 ,
so 4,900,000,000 = 4.9 × 109 in Scientific Notation
So the number is written in two parts:
Just the digits (with the decimal point placed after the first digit), followed by
× 10 to a power that puts the decimal point where it should be
(i.e. it shows how many places to move the decimal point).
In this example, 5326.6 is written as 5.3266 × 103,
because 5326.6 = 5.3266 × 1000 = 5.3266 × 103
To figure out the power of 10, think "how many places do I move the decimal point?"
When the number is 10 or greater, the decimal point has to move to the left, and the power
of 10 is positive.
When the number is smaller than 1, the decimal point has to move to the right, so the power of 10 is
negative.
Example: 0.0055 is written 5.5 × 10-3
Because 0.0055 = 5.5 × 0.001 = 5.5 × 10-3
Example: What is 1.35 × 104 ?
You can calculate it as: 1.35 x (10 × 10 × 10 × 10) = 1.35 x 10,000 = 13,500
But it is easier to think "move the decimal point 4 places to the right" like this:
1.35
13.5
135.
1350.
13500.
Example: What is 7.1 × 10-3 ?
Well, it is really 7.1 x (1/10 × 1/10 × 1/10) = 7.1 × 0.001 = 0.0071
But it is easier to think "move the decimal point 3 places to the left" like this:
7.1
0.71
0.071
0.0071
MULIPLYING and DIVIDING with SCIENTIFIC NOTATION
What is
( 1.13 × 10-7) × (3.34 × 1022) ?
This might seem like a crazy problem, but actually we can use commutative and associative properties
of multiplication to simplify this.
This expression can be rewritten as:
1.13 × 3.34 × 10-7 × 1022
3.7742 × 10-7+22
3.7742 × 1015
Division would just be dividing the real numbers and subtracting the exponents.
Laws of Exponents
Here are the Laws (explanations follow):
Law
x1 = x
x0 = 1
x-1 = 1/x
xmxn = xm+n
xm/xn = xm-n
(xm)n = xmn
(xy)n = xnyn
(x/y)n = xn/yn
x-n = 1/xn
And the law about Fractional Exponents:
Example
61 = 6
70 = 1
4-1 = 1/4
x2x3 = x2+3 = x5
x6/x2 = x6-2 = x4
(x2)3 = x2×3 = x6
(xy)3 = x3y3
(x/y)2 = x2 / y2
x-3 = 1/x3