Chabot Mathematics
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Chabot Mathematics. §1.6 Exponent Properties. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected] MTH 55. 1.5. Review §. Any QUESTIONS About §1.5 → (Word) Problem Solving Any QUESTIONS About HomeWork §1.5 → HW-01. Exponent PRODUCT Rule.

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Bruce mayer pe licensed electrical mechanical engineer bmayer chabotcollege

Chabot Mathematics

§1.6 ExponentProperties

Bruce Mayer, PE

Licensed Electrical & Mechanical [email protected]


Review

MTH 55

1.5

Review §

  • Any QUESTIONS About

    • §1.5 → (Word) Problem Solving

  • Any QUESTIONS About HomeWork

    • §1.5 → HW-01


Exponent product rule

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


Quick test of product rule

Quick Test of Product Rule

  • Test


Example product rule

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) x3x5b) 62 67  63

    c) (x + y)6(x + y)9d) (w3z4)(w3z7)


Example product rule1

Example  Product Rule

  • Solution a) x3x5 = 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

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


Quick test of quotient rule

Quick Test of Quotient Rule

  • Test


Example quotient rule

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 rule1

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

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


Example the exponent zero

Example  The Exponent Zero

  • Simplify: a) 12450b) (−3)0c) (4w)0d) (−1)80e) −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

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


Quick test of power rule

Quick Test of Power Rule

  • Test


Example power rule

Example  Power Rule

  • Simplify: a) (x3)4b) (42)8

  • Solution a) (x3)4= x34

    = x12

  • Solution b) (42)8= 428

    = 416

Base is x

Base is 4


Raising a product to a power

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


Quick test of product to power

Quick Test of Product to Power

  • Test


Example product to 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

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


Quick test of quotient to power

Quick Test of Quotient to Power

  • Test


Example quotient to a power

Example  Quotient to a Power

  • Simplify: a) b) c)

  • Solution a)

  • Solution b)

  • Solution c)


Negative exponents

Negative Exponents

  • Integers as Negative Exponents


Negative exponents1

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

Example  Negative Exponents

  • Express using POSITIVE exponents, and, if possible, simplify.

    a) m–5b) 5–2c) (−4)−2d) xy–1

  • SOLUTION

    a) m–5 =

b) 5–2 =


Example negative exponents1

Example  Negative Exponents

  • Express using POSITIVE exponents, and, if possible, simplify.

    a) m–5b) 5–2c) (−4)−2d) xy−1

  • SOLUTION

    c) (−4)−2=

d) xy–1 =

  • Remember PEMDAS


More examples

More Examples

  • Simplify. Do NOT use NEGATIVE exponents in the answer.a) b) (x4)3c) (3a2b4)3d)e) f)

  • Solution

    a)


More examples1

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

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

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

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


Examples flippers1

Examples  Flippers

  • Simplify

  • SOLUTION


Summary exponent properties

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

WhiteBoard Work

  • Problems From §1.6 Exercise Set

    • 14, 24, 52, 70, 84, 92, 112, 130

  • Base & Exponent →Which is Which?


All done for today

All Done for Today

AstronomicalUnit

(AU)


Bruce mayer pe licensed electrical mechanical engineer bmayer chabotcollege

Chabot Mathematics

Appendix

Bruce Mayer, PE

Licensed Electrical & Mechanical [email protected]


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