Exponents and Scientific Notation

MATH 017

Intermediate Algebra

S. Rook

2

Overview

• Section 5.1 in the textbook– Product rule for exponents– Expressions raised to the 0 power– Quotient rule for exponents– Expressions raised to negative powers– Scientific notation

Product Rule

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Product Rule

• Consider x4 ∙ x5

x∙x∙x∙x ∙ x∙x∙x∙x∙x

x9

• Product Rule: xa ∙ xb = xa+b – When multiplying LIKE BASES, add the

exponents– Only applies when the operation is

multiplication

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Product Rule (Example)

Ex 1: Simplify: (4xy2)(2x2y3)

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Product Rule (Example)

Ex 2: Simplify: (-x2y5z)(7x4z3)

Expressions Raised to the 0 Power

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Expressions Raised to the 0 Power

• Consider x0

– As long as x ≠ 0, x0 = 1– x can also be an expression

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Expressions Raised to the 0 Power (Example)

Ex 3: (2w)0

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Expressions Raised to the 0 Power (Example)

Ex 4: -(x2y3z2)0

Quotient Rule

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Quotient Rule

• Consider x5 / x2

x∙x∙x∙x∙x / x∙x

x3

• Quotient Rule: xa / xb = xa-b – When dividing LIKE BASES, subtract the

exponents– Only applies when the operation is division

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Quotient Rule (Example)

Ex 5: Simplify:

zyx

zyx23

243

48

40

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Quotient Rule (Example)

Ex 6: Simplify:

24

78

28

16

ts

trs

Expressions with Negative Exponents

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Expressions with Negative Exponents

• Consider x2 / x6

x-4 by the quotient rulex∙x / x∙x∙x∙x∙x∙x1 / x4

• We NEVER leave an expression with a negative exponent

• Flipping an exponent and its base from the numerator into the denominator (or vice versa) reverses the sign of the exponent

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Expressions with Negative Exponents (Continued)

• x-4 = x-4 / 1 = 1 / x4

• 2-3 = 2-3 / 1 = 1 / 23 = 1 / 8 ≠ -8– The sign of the exponent DOES NOT affect

the sign of the coefficient (or base)– Whenever using the quotient rule, the initial

result goes into the numerator

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Expressions with Negative Exponents (Example)

Ex 7: Simplify – leave NO negative exponents:

342

242

12

2

tsr

tsr

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Expressions with Negative Exponents (Example)

Ex 8: Simplify – leave NO negative exponents:

yx

yx2

43

14

4

Scientific Notation

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Scientific Notation

• Scientific Notation: any number in the form of a x 10b where -10 < a < 10, a ≠ 0 and b is an integer– One non-zero number to the left of the

decimal point – the rest to the right– Count how many places and in which

direction the decimal is moved• If to the left, b is positive• If to the right, b is negative

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Scientific Notation (Example)

Ex 9: Write in scientific notation:

0.000135

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Scientific Notation (Example)

Ex 10: Write in scientific notation:

451,000

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Standard Notation

• Standard Notation: writing a number with a product of a power of ten without the power of ten– Take the decimal and move it:

• To the right if b is positive• To the left if b is negative• Fill in empty spots with zeros

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Standard Notation (Example)

Ex 11: Write in standard notation:

1.155 x 104

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Standard Notation (Example)

Ex 12: Write in standard notation:

29.3 x 10-3

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Summary

• After studying these slides, you should know how to do the following:– Apply the product rule when multiplying like bases– Evaluate expressions raised to the 0 power– Apply the quotient rule when dividing like bases– Simplify expressions raised to negative powers– Convert back and forth between scientific and

standard notation