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 The shortcut tells us to add the exponents.

Example: (3x2) (4x3)

= (3 ∙ 4)(x2 ∙ x3)

= 12x5

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: Proof that this works:

= a5-3

= a2 = a2

3

5

a

a3

5

a

a

aaaaaaaa

******

DIVIDING EXPONENTS EXAMPLES

Example: Use the quotient rule of exponents to simplify

= 57 - 5 Shortcut.

= 52

Example: Use the quotient rule of exponents to simplify

= a12 - 11 Shortcut.

= a1

= a

5

7

5

5

11

12

a

a

DIVIDING EXPONENTS

Bottom Exponent is Bigger: If the exponent on bottom is bigger, we

subtract and keep the answer on the bottom.

Example:

Proof:

5

2

7

7

257

1

37

1 7*7*7*7*7

7*7

37

1

MORE DIVISION EXAMPLES

Example: Use the quotient rule of exponents to simplify

Since the bottom exponent is larger, we subtract on bottom.

Example: Use the quotient rule of exponents to simplify

Since the bottom exponent is larger, we subtract on bottom.

14

3

y

y

314

1

y

11

1

y

12

2

x

x

212

1

x

10

1

x

SAME EXPONENTS

Same Exponents If our exponents are the same, the terms will

cancel to 1.

Example:

= x5 - 5 Subtract the exponents.

= x0

= 1

5

5

x

x

SECTION 6.1 CONTINUED

Quotient Raised to a Power We raise top and bottom to the power.

Examples:

3

yx

3

3

y

x

2

52

a

2

2

5

)2( a

254 2a

2

22

5

2 a

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:

2

3y

23y

2

1y

1

2y

2x

2

1

x

2y

NEGATIVE EXPONENT EXAMPLES

a-4

= b8c1

= b8c

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

x-2 y3

1

4a

4

1

a

1

8

c

b

2

3

y

x

3

2

x

y

2

3

x

y

MORE NEGATIVE EXPONENT EXAMPLES

Only move the term(s) with a negative exponent. Leave everything else alone.

y

x 45

yx 4

5

2

45xy

a

x

ya 245

5

22

y

x

2

52

x

y

zy

x2

64

zx

y6

24

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 EXAMPLESExample: 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!

AIM: tamitacker

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The final is in MML and is due at the end of this Unit with all the other U9 Assignments.