Download Exponential Laws and Equations

Exponential Laws and Equations
The Introduction to Exponents!
A long time ago XXXX was written X X X X . This became so long
and complicated that X X X X was turned into the notation X4, which means X
multiplied by itself four times. X4 is called an exponential notation.
Exponents are represented in this form.
(n)
X
 Exponent

Base
Exponent: The number of times the base is multiplied by its self.
Base: The number that is being multiplied.
X0 = 1
X1 = X
X2 = X X
X3 = XXX
And so on…….
To change a number into a exponent:
l
log number
log new base
= New Exponent
Example:
Log27 = 3
Log3
27 can be represented as 33
33 = (3)(3)(3)
= (9)(3)
= 27
** Sometimes the operator ^ is used in calculations to represent an exponent
Example:
Xn = X^n
When using large numbers such as 1000000, they can get large and messy. We
can simplify them by using exponents.
Example: 1 000 000 = 101010101010
= 106
Negative Bases:
Negatives being applied to the bases must be watched carefully. If the negative
is not in brackets with the base the negative will also be applied to the answer.
Example:
-25 = -(22222)
= -32
If the negative being applied to the base is in brackets and the exponent is even
than the negative will not apply to the answer.
Example:
(-2)4= (-2)(-2)(-2)(-2)
= 16
If the negative being applied to the base is in brackets and the exponent is odd
than the negative will apply to the answer.
Example:
(-2)5 = (-2)(-2)(-2)(-2)(-2)
= -32
Exponent Laws!
Rule 1:
If the bases are the same and multiplied, you
can add their exponents
Rule 2:
If the bases are the same and divided, you can
subtract their exponents
Rule 3:
If a power is raised to a further power, you must
multiply the exponents
If you have different bases, multiplied with
each other, raised to a power, you must apply
rule 3 for each of the bases! Remove the ( )'s!
Rule 4:
Rule 5:
Same goes for division! The exponent goes for
the top and the bottom (the numerator and the
denominator)
Rule 6:
or
Rules and Laws Explained:
Negative exponents:
(if you are negative on top rather go
underground) or (if you buries underground get
on top and be positive
P
Rule 1: Negative Exponents
A number in exponential notation can be expressed as a reciprocal. A
reciprocal is 1 over the base to the opposite sign on the exponent.
Example:
Xn = 1/x-n
and
X^-n = 1/x^n
3-3 = 1/33
= 1/(3)(3)(3)
= 27
33 = (3)(3)(3)
= 27
Rule 2: Multiplication
If the bases are the same and you are multiplying then you can add the
exponents.
Example:
X2 + X2 = X2+2
= X4
26 + 22 = 26+2
= 28
= (2)(2)(2)(2)(2)(2)(2)(2)
= 256
Rule 3: Division
If the bases are the same and you are dividing the you can subtract the
exponents.
Example:
X / X2 = X3-2
= X1
=X
26 / 24 = 26-4
= 22
=(2)(2)
=4
Rule 4: Raising a power to an exponent
When you are raising a power to an exponent keep the base and multiply
the exponents.
Example:
(r4)5 = r4*5
= r20
(52)3 = 52*3
= 56
= 15625
Rule 5: Different Bases
If you have two bases that are being multiplied or divided and raised to a
power you have to use rule 4 for both of the bases. Then you remove the
brackets.
Example:
(Multiplying)
(XY)3 = X3 Y3
(3*4)2 = 32 42
= (3)(3) (4)(4)
= (9) (16)
= 144
(Dividing)
(X/Y)2 = X2 / Y2
(4/2)2 = 42/ 22
= (4)(4) / (2)(2)
= 16 / 4
=4
Questions To Try:
1.
2.
3.
 64x6y9
25x6y12
416x4y12
3
–5a-3 / a2
5. 6x12y3z2 / -6x6z2
6. (-2a3b2)2 / 4a3b2
7. –(-xy)4
8. (17x5y9z8)2
9. (xa)a+5
10. 23x2y3(22x4y3)(2xy2z)
11. 32(x4)(y3)(x8)(y2)(22)
12. x18y9z10 – x6z5
13. 2xy3(4x2y) / (2xy2)3
14. 4a3b4c6 / 12a2b4c0
15.
(-5x5y9)2(-5x8y2)3
4.
Answers:
1. 3 64x6y9
(43x6y9)1/3
4x2y9
The root becomes the denominator. To take the cubed root of am exponent,
divide by 3.
2. 25x6y12
(52x6y12)1/2
5x2y6
The root becomes the denominator. To take the square root of an exponent,
divide by 2.
3. 464x4y12
(43x4y12)1/4
43/4xy3
The root becomes the denominator. To take the fourth root of an exponent divide
by 2.
4. –5a-3 / a2
-5a-3-2
-5a-5
-5 / a5
The base is the same so subtract the exponents. The negative exponent
becomes positive by flipping the base.
5. 6x12y3z2 / -6x6z2
-6x12-6y3z2-2
-6x6y3
A (+) divided by a (–) equals a (-). The exponents on the like bases are
subtracted.
6. (-2a3b2)2 / 4a3b2
4a6b4 / 4a3b2
4a6-3 b4-2
4a3b2
The terms in the brackets are squared. The exponents on the bases that are the
same are subtracted.
7. –(-xy)4
-(-x4y4)
x4y4
The brackets are removed by multiplying the exponent on the terms by the
exponent on the bracket. A (-) times a (-) is a (+)
8. (17x5y9z8)2
289x10y18z16
The brackets are removed by multiplying the exponents on the terms by the one
on the bracket.
9. (xa)a+5
xa2+5a
A*A=A2
5*A = 5A
10. 23x2y3(22x4y3)(2xy2z)
8(4)(2)x2+4+1y3+8+2z
64x7y8z
The numbers are multiplied by themselves. The terms are being multiplied so
exponents of the same base are added.
Test Questions:
–(-x2y3)6
(xaya) a+1
16a4b6c8
x7y3z2  x3y3z2
–6a -4
a2
6. 23x2y2 ( 22x4y3)( 2xy2z)
7. 2xy3(4x2y)
(2xy2)3
8. 4a3b4c6
12a2b4c0
9. 3 27x6y12
10. (3x6y9)2 – (6x6y9)2
1.
2.
3.
4.
5.
Test Answers:
1. –(-x2y3)6
= -(-x12y18)
= x12y18
2. (xaya)a+6
= x a2+6a y a2+6a
3. 16a4b6c8
= (24a4b6c8) ½
= 4a2b3c4
4. x7y3z2  x3y3z2
= x 7-3 y 3-2 z 2-2
= x4 y
5.
–6a-4
a2
= -6a-4-2
=-6a-6
= -6
a6
6. 23x2y3 (22x4y3)( 2xy2z)
= 8(4)(2) x 2+4+1 y 2+3+3 z
= 64 x7y8z
7. 2xy2(4x2y)
(2xy2)3
= 8x3y4
23x3y6
=8-8 x 3-3 y 4-6
= y –2
=1
y2
8. 4a3b4c6
12a2b4c0
= 1/3 a 3-2 b 4-4 c 6-0
= 1/3 ac6
= ac6
3
9. 3 27x6y12
= (33 x6y12) 1/3
= 3x2y6
10. (3x6y9)2 – (6x6y9)2
= 32x12y18 – 62x12y18
=9-36 x 12-12 y 18-18
= -27
= 1
27
Links To Exponent WebPages:
1) http://www.sosmath.com/algebra/logs/log3/log3.html
2) http://www.math.com/school/subject2/lessons/S2U2L2GL.html
3) http://www.edhelper.com/exponents.htm
Bibliography:
Algebra Basics, “Exponents”, 2000,
http://www.math.com/school/subject2/lessons/S2U2L2GL.html
Ed Helper.com, “Exponent Worksheets”
http://www.edhelper.com/exponents.htm
“Exponential and Surd Worksheet Summary”
http://www.easymaths.org/Gr%2011/Algebra/exsitemap.htm
Henderson, Douglas, Starting Points In Math 10, 1982, Canada: Ginn
and Company Educational Publishers
Marcus, Nancy, “Exponential Rules, 2002, USA: Math Medics
http://www.sosmath.com/algebra/logs/log3/log3.html
Math League Multimedia, “Decimals, Whole Numbers, and
Exponents”, 2002,
http://www.mathleague.com/help/decwholeexp/decwholeexp.htm