Transcript Document

Exponents and Scientific
Notation
MATH 017
Intermediate Algebra
S. Rook
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
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Product Rule
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)
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Expressions Raised to the 0
Power
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
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Quotient Rule
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:
40 x 3 y 4 z 2
48 x 3 y 2 z
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Quotient Rule (Example)
Ex 6: Simplify:
16rs 8t 7
28s 4t 2
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Expressions with Negative
Exponents
Expressions with Negative
Exponents
• Consider x2 / x6
x-4 by the quotient rule
x∙x / x∙x∙x∙x∙x∙x
1 / 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:  2r 2 s 4t 2
 2 4 3
12r s t
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Expressions with Negative
Exponents (Example)
Ex 8: Simplify – leave NO negative
exponents:
3 4
4x y
 14 x  2 y
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Scientific Notation
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
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