Download Chapter 4 Study Guide

6th Grade A&E Study Guide
Chapter 4 – Number Theory
Properties of Exponents – know these properties and why they work
an . am = an+m
am
 amn
an
(am)n = amn
(am)n = anmn
a1 = a
a0 = 1 (where a ≠ 0)
1
a-1 =
(where a ≠ 0)
a
1
a-n = n (where a ≠ 0)
a
Multiplying Like Bases:
Dividing Like Bases:
Power to a Power:
Product to a Power:
Power of 1:
Power of 0:
Negative Exponents:
Watch out …
4x2 ≠ 16x2 … the exponent ‘2’ applies to x only since ‘x’ is the base
(4x)2 = 16x2 … here the exponent does apply to ‘4’ also since ‘4x’ is the base
-62 … read as “the opposite of six squared” = -(6)(6) = -36
(-6)2 … read as “negative six squared” = (-6)(-6) = 36
23 + 24 ≠ 27 … exponents are added when multiplying (not adding) like bases
a)
m12 (m5) _____
b)
e)
(7n3)2 ______
f)
y9
_____
y5
60 _____
c)
-34 _____ g)
d)
(x5)6(x) _____ h)
Evaluating Algebraic Expressions With Exponents
Substitute with parentheses and evaluate using Order of Operations.
Ex:
i)
Evaluate -y2 – x + x5
-(5)2 – (-2) + (-2)5
-25 + 2 + -32
-23 – 32
-55
2
2
m = -3 ; n = -4 ; p = 6
n + m – pm - mn
x = -2 ; y = 5
9-2_____
3
______
g 4
Simplifying Exponent Expressions
Ex:
m5 n8
n7
5-8 8-1
-3 7
=
m
n
=
m
n
=
m8 n
m3
[notice: no negative exponents in final answer]
Another way to think of it is 5 m’s on top vs. 8 m’s on bottom = 3 m’s on bottom
8 n’s on top vs. 1 n on bottom = 7 n’s on top
Ex:
6 x 4 y 2
The negative in an exponent indicates reciprocal so this equals …
3 x 5 y 6
6x5 y 2 y 6
= 2xy8
3x 4
Evaluate the following.
j)
e12 f
e6 f 4 g
_________
k)
w 2 x 7
w 5 x 3
_________
l)
12m 8 n 3 q
= ________
18m 3 n 9 q
Scientific Notation
Used to represent extremely large or extremely small numbers, scientific notation
has the form:
a x 10n where 1 ≤ a < 10 and n is an integer
Standard form and Scientific Notation:
4,123,000 = 4.123 x 106
0.000000582 = 5.82 x 10-7
As you see the power of 10 indicates how far the decimal point moves
The sign of the exponent indicates direction:
Positive exponents used for numbers ≥ 10
Negative exponents used for numbers < 1
Operating with Scientific Notation:
Ex: (4.25 x 1030) (3 x 1022)
(4.25 x 3) (1030 x 1022)
12.75 x 1052
.
÷10
10
1.275 x 1053
Ex:
m)
writing these in standard form would take too
long so use properties to make it easier …
Notice this is not in scientific notation so …
Div. by 10 and mult by 10 simultaneously
1.5 x 10 15
1.5 10 15
=
x
= 0.25 x 10-33 = 2.5 x 10-34
18
18
6
6 x 10
10
4 (3.8 x 10-5)
n)
(9.4 x 10-41) (3.1 x 10-12)
o)
37.5 x 10 62
0.25 x 10 9
Prime Factorization
m is a factor of n if m divides n evenly. Ex: 4 is a factor of 12
A prime number is a whole number with exactly 2 factors (1 and itself). Ex: 3, 47
A composite number is a whole number with more than 2 factors. Ex: 20, 51
When listing a number’s factors, each factor has a “partner” on the other side of
the number’s square root. The factor times the partner equals the number.
Ex: Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 [6 is the square root of 36]
since 62 = 36
A perfect square has an odd amount of factors since its square root is its own partner.
Every composite number can be written as a product of primes. This is a unique
representation of a number – sort of like a number’s “fingerprint”.
Factor trees are commonly used to find prime factorization. Remember to express
this product using exponents for multiple prime factors.
Ex:
84 = 22 . 3 . 7
Find the prime factorization:
p)
440
_____________
q)
819
_______________
GCF / LCM
There are several methods for finding GCF and LCM
Ex: GCF and LCM of 48 and 60
Prime Factorizations
48: 24 . 3
60: 22 . 3 . 5
GCF = 22 . 3 = 12
[product of common factors]
. 2 .
LCM = 12 2 5 = 240 [product of LCM and leftover factors]
or 24 . 3 . 5 = 240 [product of each base’s highest power]
Find GCF and LCM using Prime Factorization
r)
40 and 56
s)
210 and 84
t)
330 and 132
Repeated Division
Ex:
Find the GCF and LCM of 90 and 63
Find GCF and LCM.
u)
240 and 176
v)
324 and 216
w)
9 and 22
Don’t forget that with 3 numbers, a) find the GCF
b) divide factors common to any 2 numbers
w)
280
120
525
Use whichever method you’d like to find GCF and LCM.
y)
40w2x3y and 24w6y5
a)
70m5n and 48m9n3