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Compare these two cases
€
(−6)2 = −6• −6 = +36
−62 = −(6)2 = −(6) • (6) = −36
6 IS THE BASE(THE NEG. SIGN IS NOT PARTOF THE BASE, IT MUST REMAIN PART OF THE ANSWER)
-6 IS THE BASE
Evaluating Expression with Exponents
Evaluate 2x2(x+y)When x=6 & y=3
1. Put in x & y values
2x2(x+y)= 2(6)2(6+3)
Evaluating Expression with Exponents
Evaluate 2x2(x+y)When x=6 & y=3
1. Put in x & y values
2. Use PEMDAS2x2(x+y)= 2(6)2(6+3)= 2(6)2(9)= 2(36)9= 72•9= 648
MULTIPLYING SIMILAR BASES
THE RULE IS TO ADD THE EXPONENTS
€
X 2 • X 3 = X 2+3 = X 5
X a • X b = X a +b
DIVIDING SIMILAR BASES
€
X 5
X 2=
X • X • X • X • X
X • X= X • X • X = X 3
THE RULE IS TO SUBTRACT THE EXPONENT OF
THE DENOMINATOR FROM THE EXPONENT OF
THE NUMERATOR
DIVIDING SIMILAR BASES
€
X a
X b= X a−b
X 6
X 2= X 6−2 = X 4
X 7
X−5= X 7−(−5) = X12
THE RULE IS TO SUBTRACT THE EXPONENTS
NEGATIVE EXPONENTS
€
Y −4 =1
Y 4
1
Y −4= Y 4
€
X−2 =1
X 2
1
X−2= X 2
You can change a negative exponent to positive by switching it’s base from numerator to denominator or vice versa.
NEGATIVE EXPONENTS
€
X−2 =1
X 2
MOVE THE BASE & EXPONENT FROM THE NUMERATOR TO THE
DENOMINATOR OR VICE VERSA AND CHANGE THE SIGN OF THE EXPONENT
€
X−2 X 2
X 2
⎛
⎝ ⎜
⎞
⎠ ⎟=
X−2 • X 2
1• X 2
MULTIPLY NUMERATORS & MULTIPLY DENOMINATORS
NEGATIVE EXPONENTS
€
X−2 X 2
X 2
⎛
⎝ ⎜
⎞
⎠ ⎟=
X−2 • X 2
1• X 2=
X−2+2
X 2=
X 0
X 2
ANYTHING TO THE ZERO POWER IS EQUAL TO ?
NEGATIVE EXPONENTS
Check Out These Patterns
€
23 = 8
22 = 4 = (8 ÷ 2)
21 = 2 = (4 ÷ 2)
20 =1 = (2 ÷ 2)
2−1 =1
2= (1÷ 2)
€
53 =125
52 = 25 = (125 ÷ 5)
51 = 5 = (25 ÷ 5)
50 =1= (5 ÷ 5)
5−1 =1
5= (1÷ 5)
Or For Anything
€
x 3 = x • x • x = x 4
x( )
x 2 = x • x = x 3
x( )
x1 = x = x 2
x( )
x 0 =1= xx( )
x−1 =1
x= 1
x( )
€
X−2 X 2
X 2
⎛
⎝ ⎜
⎞
⎠ ⎟=
X−2 • X 2
1• X 2=
X−2+2
X 2=
X 0
X 2=
1
X 2
THIS IS WHY
SWITCHING A NEGATIVE EXPONENT CHANGES ITS SIGN
Let’s compare the Neg. ExponentRule with the Dividing Fraction Rule
To divide fractions you
INVERT THE 2ND FRACTION AND CHANGE THE DIVISION SIGN TO MULTIPICATION
For negative exponents you
INVERT THE BASE WITH AND CHANGE THE SIGN OF THE EXPONENT
What if the negative exponent is in the denominator?
€
1
x−2=
The same rule of inverting the base with the exponent and making the exponent positive applies but let see why this is so.
What if the negative exponent is in the denominator?
€
1
x−2=
1
x−2
x 2
x 2
⎛
⎝ ⎜
⎞
⎠ ⎟=
x 2
x−2+2=
Use the Multiplicative Identity to Simplify
What if the negative exponent is in the denominator?
€
1
x−2=
1
x−2
x 2
x 2
⎛
⎝ ⎜
⎞
⎠ ⎟=
x 2
x−2+2=
x 2
x 0=
Do you remember what x0 is equal to?
What if the negative exponent is in the denominator?
€
1
x−2=
1
x−2
x 2
x 2
⎛
⎝ ⎜
⎞
⎠ ⎟=
x 2
x−2+2=
x 2
x 0=
x 2
1= x 2
INVERT AND CHANGE EXPONENT SIGN
SUMMARY SO FAREXPONENTIAL NOTATION
€
X
2
X IS THE BASE
2 IS THE EXPONENT
€
X−2 =1
X 2
1
X−2= X 2
DIVIDING EXPONENTS
SUBTRACT THE EXPONENT OF THE DENOMINATOR FROM THE
EXPONENT OF THE NUMERATOR
MULTIPLYING EXPONENTS
ADD THE EXPONENTS
€
X A • X B = X A +B
€
X A
X B= X A−B
INVERTING A NEGATIVE EXPONENT CHANGES ITS SIGN
POWER TO A POWER VS. MULT. SIMILAR BASES
€
X 2 • X 3 = X 2+3 = X 5
X 2( )
3= X 2
( ) X 2( ) X 2
( ) = X 2+2+2 = X 6
POWER TO A POWER
€
X 2( )
3= X 2
( ) X 2( ) X 2
( )
= X • X( ) X • X( ) X • X( ) = X 6
MULTIPLY THE EXPONENTS
€
X a( )
b= X a⋅b
NOTHING REALLY(A MONOMIAL IS A PRODUCT)
A monomial is a term with a number and one or more variables (letters) raised to some power. Examples are:
€
5x
a
−23a5
2,345m2n−5 p23
€
AB( )C
= AC BC
€
AB( )C
= A1B1( )
C= A1•C B1•C = AC BC
Let’s take a close look at DISTRIBUTING THE POWER
Simplify (4d5)2
= 41•2 • d5•2
= 42 d10
1. Multiply the outsidepower to the insidepowers.
Powerof a Monomial
Simplify (4d5)2
= 41•2 • d5•2
= 42 d10
= 16 d10
1. Multiply the outsidepower to the insidepowers.
2. Simplify
Powerof a Monomial
Power of a MonomialSimplify (2x3y4)5
= 21•5 • x3•5•y4•5
= 25 x15y20
or 32 x15y20
1. Multiply the outsidepower to the insidepowers.
2. Simplify
Take a Powerof a QuotientSimplify
1. Multiply the outsidepower to the insidepowers.
€
3x 3y
7z2
⎛
⎝ ⎜
⎞
⎠ ⎟
4
€
3x 3y
2z2
⎛
⎝ ⎜
⎞
⎠ ⎟
4
=34 x 3•4 y1•4
24 z2•4
=34 x12y 4
24 z8=
Take a Powerof a QuotientSimplify
1. Multiply the outsidepower to the insidepowers.
2. Simplify
€
3x 3y
7z2
⎛
⎝ ⎜
⎞
⎠ ⎟
4
€
3x 3y
2z2
⎛
⎝ ⎜
⎞
⎠ ⎟
4
=34 x 3•4 y1•4
24 z2•4
=34 x12y 4
24 z8=
81x12y 4
16z8
Mult/Divide Monomials with
Exponents
1. Multiply the numbers2. Multiply the Similar
Variables by adding the exponents.
€
3x 3y 5( ) • −8x 2y( )
€
3x 3y 5( ) • −8x 2y( )
= 3 • −8x 3+2y 5+1
Mult/Divide Monomials with
Exponents
1. Multiply the numbers2. Multiply the Similar
Variables by adding the exponents.
3. Simplify
€
3x 3y 5( ) • −8x 2y( )
€
3x 3y 5( ) • −8x 2y( )
= 3 • −8x 3+2y 5+1
= −24 x 5y 6
5 Exponent Rules1.When multiplying similar bases add exponents
2.When dividing similar bases subtract exponents
3.When inverting change exponents’ sign
4.When taking a power of a power multiply exponents
5.When taking a power of a product or quotient distribute the outside power to the inside powers
EXPONENTIAL NOTATION
€
X
2
X IS THE BASE
2 IS THE EXPONENT
6 IS THE BASE( . THE NEG SIGN IS NOT PART
, OF THE BASE IT MUST REMAIN )PART OF THE ANSWER
-6 IS THE BASE
Compare these two cases
€
( − 6 )
2
= − 6 • − 6 = + 36
− 6
2
= − ( 6 )
2
= − ( 6 ) • ( 6 ) = − 36
INVERTING A NEGATIVEEXPONENTS CHANGES ITS SIGN
€
X
− 2
=
1
X
2
€
1
X−2= X 2
DIVIDING EXPONENTS
SUBTRACT THE EXPONENTS
MULTIPLYING EXPONENTS
ADD THE EXPONENTS
€
X A • X B = X A +B
€
X A
X B= X A−B
POWER TO A POWER
€
X
2
( )
3
= X
2
( )X
2
( )X
2
( )= X
2 + 2 + 2
= X
6
THIS IS APOWERFULIDEA
Anything to the zero power
equals 1
Take A PowerTo A PowerSimplify
1. Multiply the outsidepower to the insidepowers.
2. Simplify
€
3 x3
y
7 z2
⎛
⎝
⎜
⎞
⎠
⎟
4
€
3 x3
y
2 z2
⎛
⎝
⎜
⎞
⎠
⎟
4
=3
4
x3 • 4
y1 • 4
24
z2 • 4
=3
4
x12
y4
24
z8
=81 x
12
y4
16 z8
EXPONENTS (Or POWERS are REPEATED MULTIPICATION
5 Exponent Rules1.When multiplying similar bases add exponents
2.When dividing similar bases subtract exponents
3.When inverting change exponents’ sign
4.When taking a power of a power multiply exponents
5.When taking a power of a product or quotientdistribute the outside power to the inside powers
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