2. 7(x + 3)= 105 7x + 21 = 105 -21 -21 7x = 84 7 7 x = 12 solve. 1. 5x + 18 = -3x – 14 +3x +3x 8x...
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Bell Ringer
2. 7(x + 3)= 105 7x + 21 = 105 -21 -21 7x = 84 7 7 x = 12
Solve.1. 5x + 18 = -3x – 14 +3x +3x 8x + 18 = -14
- 18 -18 8x = -32
8 8 x = -4
Quiz Results
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Exponents and Radicals
NCP 503: Work with numerical factorsNCP 505: Work with squares and square roots of numbersNCP 506: Work problems involving positive integer exponents*NCP 504: Work with scientific notationNCP 507: Work with cubes and cube roots of numbersNCP 604: Apply rules of exponents
Basic Terminology
34
Exponent
Base
= 3•3•3•3 = 81
The base is multiplied by itself the same number of times as the exponent calls for.
Its read, “Three to the fourth power.”
Important Examples
-34 = –(3•3•3•3) = -81
(-3)4 = (-3)•(-3)•(-3)•(-3) = 81
(-3)3 = (-3)•(-3)•(-3) = -27
-33 = –(3•3•3) = -27
Variable Expressions
x4 = x • x • x • x y3 = y • y • y
Evaluate each expression if x = 2 and y = 5
x4 y2 = (2•2•2•2)•(5•5) = 4003xy3 = 3•2•(5•5•5)
= 750
Zero Exponent PropertyNegative Exponent Property
Product of PowersQuotient of Powers
Laws of Exponents, Pt. I
Zero Exponent Property
Any number or variable raised to the zero power is 1.
x0 = 1 y0 = 1 z0 = 1
70 = 1 -540 = 1 1230 = 1
Negative Exponent
Any number raised to a negative exponent is the reciprocal of the number.
x-1 = y-1 = 5-1 =
x-2 = 3-2 = = 5-3 = =
1X
1y
15
1X2
132
19
153
1 .125
Negative Exponent
3x-3 = 5y-2 =
2x-2 y2= 3-2 x4=
3x3
5y2
2y2
x2 x4
32
Only x is raised to the -3 power!
Only x is on the bottom.
x4
9 =
Product of Powers
This property is used to combine 2 or more exponential expressions with the SAME base.
53•52 = (5•5•5)•(5•5)
= 55
If the bases are the same, add the exponent!
x4•x3 = (x•x•x•x)•(x•x•x) = x7
Multiplication NOT Addition!
Product of Powers
6-2•6-3
=
165
162•63
=
=
17776
x-5•x-7
=
1x12
1x5•x7
=
n-3•n5 =
n2 n-3+5 =
Product of powers also work with negative exponents!
Quotient of Powers
This property is used when dividing two or more exponential expressions with the
same base.
x6
x3= x6-
3
= x3
Subtract the exponents! (Top minus the bottom!)
Quotient of Powers
67
65= 67-
5
= 62
x3
x5= x3-
5
= x-2 =
1x2
x3
x5= x ∙ x ∙
xx∙x∙x∙x∙x
=OR
1x2
= 36
Power of a Power
This property is used to write an exponential expression as a single power of the base.
(63)4 = 63•63•63•63
= 612
When you have an exponent raised to an exponent, multiply the
exponents!
(x5)3 = x5•x5•x5 = x15
Power of a Product
(xy)3
(2x)5
(xyz)4
Power of a Product – Distribute the exponent on the outside of the parentheses to all of the terms inside of the parentheses.
= x3y3
= 25 ∙ x5
= 32x5
= x4 y4 z4
Power of a Product
(x3y2)3
(3x2)4
(3xy)2
More examples…
= x9y6
= 34 ∙ x8
= 81x8
= 32 ∙ x2 ∙ y2
= 9x2y2
Power of a Quotient
Power of a Quotient – Distribute the exponent on the outside of the parentheses to the numerator and the denominator of the fraction.
xy
=( )5 x5
y5( )
More Examples
43 72 aa 4372 a 714a
232 285 rrr 232285 r 780r
33xy 3333 yx 3327 yx
2
3
2
b
a
22
22
3
2
b
a2
2
9
4
b
a
3522 nm 3532312 nm 15632 nm 1568 nm
x
x
2
8 4
12
8 14x 34x
5
3
3
9
z
z 35
1
3
9
z
2
13x 2
3
x
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