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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected]PowerPoint Presentation

Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected]

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1.5

Review §- Any QUESTIONS About
- §1.5 → (Word) Problem Solving

- Any QUESTIONS About HomeWork
- §1.5 → HW-01

Exponent PRODUCT Rule

- For any number a and any positive integers m and n,

Exponent

Base

- In other Words: To MULTIPLY powers with the same base, keep the base and ADD the exponents

Example Product Rule

- Multiply and simplify each of the following. (Here “simplify” means express the product as one base to a power whenever possible.)a) x3x5 b) 62 67 63
c) (x + y)6(x + y)9 d) (w3z4)(w3z7)

Example Product Rule

- Solution a) x3x5 = x3+5Adding exponents
= x8

- Solution b) 62 67 63 = 62+7+3
= 612

- Solution c) (x + y)6(x + y)9 = (x + y)6+9
= (x + y)15

- Solution d) (w3z4)(w3z7) = w3z4w3z7
= w3w3z4z7

= w6z11

Base is x

Base is 6

Base is (x + y)

TWO Bases: w & z

Exponent QUOTIENT Rule

- For any nonzero number a and any positive integers m & n for which m > n,

- In other Words: To DIVIDE powers with the same base, SUBTRACT the exponent of the denominator from the exponent of the numerator

Example Quotient Rule

- Divide and simplify each of the following. (Here “simplify” means express the product as one base to a power whenever possible.)
- a) b)

- c) d)

Example Quotient Rule

- Solution a)

Base is x

- Solution b)

Base is 8

- Solution c)

Base is (6y)

- Solution d)

TWO Bases: r & t

The Exponent Zero

- For any number a where a≠ 0

- In other Words: Any nonzero number raised to the 0 power is 1
- Remember the base can be ANY Number
- 0.00073, 19.19, −86, 1000000, anything

- Remember the base can be ANY Number

Example The Exponent Zero

- Simplify: a) 12450 b) (−3)0c) (4w)0 d) (−1)80 e) −80
- Solutions
- 12450 = 1
- (−3)0 = 1
- (4w)0 = 1, for any w 0.
- (−1)80 = (−1)1 = −1
- −80 is read “the opposite of 80” and is equivalent to (−1)80: −80 = (−1)80= (−1)1 = −1

The POWER Rule

- For any number a and any whole numbers m and n

- In other Words: To RAISE a POWER to a POWER, MULTIPLY the exponents and leave the base unchanged

Example Power Rule

- Simplify: a) (x3)4 b) (42)8
- Solution a) (x3)4= x34
= x12

- Solution b) (42)8= 428
= 416

Base is x

Base is 4

Raising a Product to a Power

- For any numbers a and b and any whole number n,

- In other Words: To RAISE A PRODUCT to a POWER, RAISE Each Factor to that POWER

Example Product to Power

- Simplify: a) (3x)4 b) (−2x3)2 c) (a2b3)7(a4b5)
- Solutions
- (3x)4 = 34x4 = 81x4
- (−2x3)2= (−2)2(x3)2 = (−1)2(2)2(x3)2 = 4x6
- (a2b3)7(a4b5) = (a2)7(b3)7a4b5
= a14b21a4b5Multiplying exponents

= a18b26 Adding exponents

Raising a Quotient to a Power

- For any real numbers a and b, b ≠ 0, and any whole number n

- In other Words: To Raise a Quotient to a power, raise BOTH the numerator & denominator to the power

Negative Exponents

- Integers as Negative Exponents

Negative Exponents

- For any real number a that is nonzero and any integer n

- The numbers a−n and an are thus RECIPROCALS of each other

Example Negative Exponents

- Express using POSITIVE exponents, and, if possible, simplify.
a) m–5b) 5–2 c) (−4)−2 d) xy–1

- SOLUTION
a) m–5 =

b) 5–2 =

Example Negative Exponents

- Express using POSITIVE exponents, and, if possible, simplify.
a) m–5 b) 5–2c) (−4)−2d) xy−1

- SOLUTION
c) (−4)−2=

d) xy–1 =

- Remember PEMDAS

More Examples

- Simplify. Do NOT use NEGATIVE exponents in the answer.a) b) (x4)3 c) (3a2b4)3d) e) f)

- Solution
a)

More Examples

- Solution
b) (x−4)−3 = x(−4)(−3) = x12

c) (3a2b−4)3 = 33(a2)3(b−4)3

= 27 a6b−12 =

d)

e)

f)

Factors & Negative Exponents

- For any nonzero real numbers a and b and any integers m and n

- A factor can be moved to the other side of the fraction bar if the sign of the exponent is changed

Examples Flippers

- Simplify

- SOLUTION
- We can move the negative factors to the other side of the fraction bar if we change the sign of each exponent.

Reciprocals & Negative Exponents

- For any nonzero real numbers a and b and any integer n

- Any base to a power is equal to the reciprocal of the base raised to the opposite power

Summary – Exponent Properties

This summary assumes that no denominators are 0 and that 00 is not considered. For any integers m and n

WhiteBoard Work

- Problems From §1.6 Exercise Set
- 14, 24, 52, 70, 84, 92, 112, 130

- Base & Exponent →Which is Which?

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