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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
4
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
5
Product Rule (Example)
Ex 1: Simplify: (4xy2)(2x2y3)
6
Product Rule (Example)
Ex 2: Simplify: (-x2y5z)(7x4z3)
Expressions Raised to the 0 Power
8
Expressions Raised to the 0 Power
• Consider x0
– As long as x ≠ 0, x0 = 1– x can also be an expression
9
Expressions Raised to the 0 Power (Example)
Ex 3: (2w)0
10
Expressions Raised to the 0 Power (Example)
Ex 4: -(x2y3z2)0
Quotient Rule
12
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
13
Quotient Rule (Example)
Ex 5: Simplify:
zyx
zyx23
243
48
40
14
Quotient Rule (Example)
Ex 6: Simplify:
24
78
28
16
ts
trs
Expressions with Negative Exponents
16
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
17
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
18
Expressions with Negative Exponents (Example)
Ex 7: Simplify – leave NO negative exponents:
342
242
12
2
tsr
tsr
19
Expressions with Negative Exponents (Example)
Ex 8: Simplify – leave NO negative exponents:
yx
yx2
43
14
4
Scientific Notation
21
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
22
Scientific Notation (Example)
Ex 9: Write in scientific notation:
0.000135
23
Scientific Notation (Example)
Ex 10: Write in scientific notation:
451,000
24
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
25
Standard Notation (Example)
Ex 11: Write in standard notation:
1.155 x 104
26
Standard Notation (Example)
Ex 12: Write in standard notation:
29.3 x 10-3
27
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