OPERATIONS WITH NUMBERS AND EXPONENTS
[Pages:20]OPERATIONS WITH NUMBERS AND EXPONENTS
1
n1 = n2
(a m )n = a m n
Unit Overview
This unit begins with a review of real numbers, their properties, and the order of operations. In addition, the various properties of integer exponents are reviewed and extended to include rational exponents. Using the property of exponents and rational exponents, expressions in radical form will be rewritten in exponential form and vice versa.
Operations with Numbers
Types of Numbers
natural numbers: whole numbers: integers:
rational numbers:
1, 2, 3, .... 0, 1, 2, 3, ... ...?2, ?1, 0, 1, 2, ... p where p and q are integers and q 0 q
Rational Numbers -- Recipes (02:54)
irrational numbers: real numbers:
numbers whose decimal part does not terminate or repeat all rational and all irrational numbers
Irrational Numbers -- Travel (02:22)
Real Numbers
Rational Numbers
Irrational Numbers 7, - 10, , e
Integers ... - 2, -1, 0, 1, 2...
Non-Integers
2 , - 3 , - 11 34 6 0.27, 5.333...
Whole Numbers
0, 1, 2, 3, 4...
Opposite of Whole Numbers
-1, - 2, - 3, - 4...
0 Natural Numbers
(Counting Numbers)
1, 2, 3, 4, 5...
Property Closure Commutative Associative Identity Inverse
Distributive
Properties of Real Numbers
Addition a + b is a real number a + b = b + a a + (b + c) = (a + b) + c a + 0 = a a + (?a) = 0
a(b + c) = ab + ac
Multiplication
ab is a real number
ab = ba
(ab)c = a(bc)
a 1 = a
a
1 a
= 1
Order of Operations
To simplify algebraic expressions you must use an order of operations. Parentheses, Exponents, Multiply, Divide, Add, Subtract "Please Excuse My Dear Aunt Sally"
*If multiplication and division are the only two operations, work the problem from left to right *If addition and subtraction are the only two operations, work the problem from left to right.
Introduction (02:08)
Example #1: Evaluate. 6 ? 3 2 2 2 4
Simple Orders--Roller Coaster Capacity (02:33)
Example #2: Evaluate.
30 ? 3 2 + 6 ? 3 30 ? 6 + 2
24 + 2 26
More Orders--Revenue (03:06)
Example #3: Evaluate.
7(12 - (8 - 3) ? 62 ) 7(12 - 5? 62 ) 7(12 - 5? 36) 7(12 -180) 7(-168) -1176
-parenthesis
-exponents -multiplication with parenthesis -subtract within parenthesis -multiply
Exponents--Around the Loop (03:02) Stop! Go to Questions #1-4 about this section, then return to continue on to the next section.
Properties of Exponents
Product property:
am an = am+n
Quotient property:
am = am-n an
Power of a power:
(am)n = amn
(add exponents) (subtract exponents) (multiply exponents)
Power of a product: Power of a quotient:
(ab)n = a nbn
a b
n
=
an bn
(exponents to each term) (exponents to each term)
* a 0 = 1 any number to the zero power is equal to one
a -1 = 1 all negative exponents have to be changed to positive exponents a
to make an expression completely simplified.
Example #1: Simplify. c -9c14 = c 5
-add exponents
Multiplying with Like Bases (01:19)
Example #2: Simplify.
(d -4 ) 6
d
-24
=
1 d 24
-multiply exponents
Example #3: Simplify. t 11 = t 6 t5
-subtract exponents
Example #4: Simplify. = z 6 z= 6-(-2) z 8 z -2
-subtract exponents
Dividing with Like Bases (01:28)
Example #5: Simplify.
(3x 2 y 3)(-2x3 y) = -6x5 y 4
-add exponents on like bases
Multiplying Expressions with Like Bases (01:53)
Example #6: Simplify.
-3b 2c 5 = -3c 3
b 7c 2
b5
-subtract exponents on like bases
Dividing Expressions with Like Bases (01:56)
Example #7: Simplify.
(-2a 5b -3 ) 2 =4a10b -6 =4a10 b6
-multiply exponents, negative exponent becomes positive if put in the denominator
Raising a Power to a Power (02:01)
Example #8: Simplify.
2x 4 y 2 -3 3x 2 y 7
-raise each term to a power of - 3 (power of a quotient)
(2x 4 y 2 ) -3 (3x 2 y 7 ) -3
2 -3 (x 4 ) -3 ( y 2 ) -3 3-3 (x 2 ) -3 ( y 7 ) -3
-raise each number and variable within parenthesis to a power of - 3 (power of a product) -multiply exponents (power of a power)
2 -3 (x -12 )( y -6 ) 3-3 (x -6 )( y -21)
-subtract exponents
2 -3 (x -6 )( y15 ) 3 -3
33 ( y15 ) 23(x6)
-write all negative exponents as positive exponents -simplify
27 y15 8x6
Raising a Power to a Power in Rational Expressions (02:54)
Example #9: Simplify. (Another way to do the previous problem)
2x 4 y 2 -3 3x 2 y 7
-invert the fraction and raise to a positive power (Note: A term raised to a negative power equals the term's reciprocal.)
3x2 y 7 3 2x 4 y 2
-raise each term to a power of 3
(3x 2 y 7 )3 (2x 4 y 2)3
-multipy exponents
27x 6 y 21 8x12 y 6
-subtract exponents
27 y15 8x 6
Stop! Go to Questions #5-12 about this section, then return to continue on to the next section.
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