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WELCOME TO THE MM204
UNIT 9 SEMINAR
SECTION 6.1: EXPONENTS

Multiplying Exponential Terms


Add the exponents.
Keep base the same.
Example:
x2 * x5
= x2+5
= x7
Proof that this works:
x2 * x5
=x*x * x*x*x*x*x
= x7
MORE EXAMPLES

If there are numbers in front, we multiply those
together:
Example: Multiply (2x5) (3x6)
= (2 ∙ 3)(x5 ∙ x6)
= 6x11
Example: (3x2) (4x3)
= (3 ∙ 4)(x2 ∙ x3)
= 12x5
The shortcut tells us to add the exponents.
POWER RULE

Power Rule:

If we have an exponent raised to another exponent,
we multiply the exponents together and keep the
base the same.
Example: Use the power rule of exponents to simplify:
(22)3.
= 22*3
= 26
= 64
Example: Use power rule of exponents to simplify: (p3)10.
= p3*10
= p30
MORE POWER RULE EXAMPLES

If we have more than one thing inside the
parentheses, we raise everything inside to the
power.
Example: Simplify the expression: (2x3)5.
= 25 * x3*5 Everything inside needs to be raised to the
fifth power.
= 25 * x15
= 32x15
Example: Simplify the expression: (4x2)3.
= 43 * x2*3
= 43 * x6
= 64x6
SECTION 6.1 CONTINUED

Dividing Exponents


If we have the same letter (base) on top and bottom, we
can combine them by subtraction.
Subtract the exponents and keep the base the same.
Example:
=
a
5
a3
a5-3
= a2
Proof that this works:

a5
a3
a *a *a *a *a
a *a *a
= a2
DIVIDING EXPONENTS EXAMPLES
Example: Use the quotient rule of exponents to simplify
= 57 - 5
Shortcut.
57
55
= 52
Example: Use the quotient rule of exponents to simplify
= a12 - 11
= a1
=a
Shortcut.
a 12
a 11
DIVIDING EXPONENTS

Bottom Exponent is Bigger:

If the exponent on bottom is bigger, we subtract and
keep the answer on the bottom.
Example:


72
75
1
7 5 2
1
Proof:

73

7 *7
7 *7 *7 *7 *7
1
73
MORE DIVISION EXAMPLES
Example: Use the quotient rule of exponents to simplify


1
y 143

y 14
Since the bottom exponent is larger, we subtract on bottom.
1
y 11
Example: Use the quotient rule of exponents to simplify

y3
x2
x 12
1
x 122
1
x 10
Since the bottom exponent is larger, we subtract on bottom.
SAME EXPONENTS

Same Exponents

If our exponents are the same, the terms will cancel
to 1.
x5
x5
Example:
= x5 - 5
= x0
=1
Subtract the exponents.
SECTION 6.1 CONTINUED

Quotient Raised to a Power

We raise top and bottom to the power.
Examples:
x

y






x3
 2a 
 
 5 
3

y3

2
(2a ) 2
52
22 a 2
52
4a 2

25
SECTION 6.2 NEGATIVE EXPONENTS

Negative Exponents





Always move negative exponents to make them positive.
If the negative exponent is on bottom, move it to the top to make it
positive.
If the negative exponent is on top, move it to the bottom to make it
positive.
Only move the term with the negative exponent.
If there’s not a “bottom” (a fraction), make one!
Examples:
1
y
x
2
y2

1
 y2

2
1
x2
3
y
2
 3y 2
NEGATIVE EXPONENT EXAMPLES
b8
a-4


a
c 1
4
1
= b8c1
1
= b8c
a4
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 3
x
y 2
-2 3
x y

y2
x3

y3
x2
MORE NEGATIVE EXPONENT EXAMPLES
5x
y

4
5a 4
xy 2
5
5a 4 y 2

x
x 4y
Only move the term(s) with a negative exponent.
Leave everything else alone.
2x
y

2
5
2y 5
x2
 4x 6
y 2z

 4y 2
x 6z
SECTION 6.2: SCIENTIFIC NOTATION

Scientific Notation
Short way to write really big or really small numbers.
 You know if your number is in scientific notation when:
 There’s only one digit to the left of the decimal.
 There’s * 10some power after the decimal part.

Example: Write 123,780 in scientific notation.
= 1.23780 * 105
I moved the decimal from the end of the number. It went
five slots to the left.
MORE SCIENTIFIC NOTATION EXAMPLES
Example: Convert 45,678 to scientific notation.
We want the decimal point to end up between the 4 and the 5. We’ll have to
move the decimal point 4 places to the left, making the power of 10 a
positive 4.
45,678
= 4.5678 * 104
Example: Convert 234,005,000 to scientific notation.
We want the decimal point to end up between the 2 and the 3. We’ll have to
move the decimal point 8 places to the left, making the power of 10 a
positive 8.
234,005,000
= 2.34005 * 108
MORE SCIENTIFIC NOTATION EXAMPLES
Example: Convert 0.0000082 to scientific notation.
The decimal point needs to end up between the 8 and the 2. In order for the decimal
point to move there, it needs to travel 6 places to the right, making the power of
10 a negative 6.
0.0000082
= 8.2 * 10-6
Example: Convert 0.000157 to scientific notation.
The decimal point needs to end up between the 1 and the 5. In order for the decimal
point to move there, it needs to travel 4 places to the right, making the power of
10 a negative 4.
0.000157
= 1.57 * 10-4
THANKS FOR PARTICIPATING!

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
Read, read, read!

Email me if you have questions.

The final is in MML and is due at the end of this
Unit with all the other U9 Assignments.