This document discusses the laws of exponents. It defines exponents and provides examples of several exponent laws including: 1) The exponent of a number indicates how many times to multiply the base by itself. 2) Any base raised to the 0th power equals 1. 3) A negative exponent indicates taking the reciprocal of the base raised to the positive value of the exponent. 4) Laws for multiplying, dividing, distributing exponents across parentheses, and operating on quotients of expressions with the same exponent.

1. Prepared by:
Jessa T. Briones
Ronnie Rey A. Bejasa
Divine Grace Gallego
BSEd – Math III

2. 
Objectives:
• I can identify the product rule of exponent in like
terms.
• I can simplify carefully the exponents of same bases
of like terms.
• I can apply the laws of exponents for integer
exponent.

3. Laws of Exponents
base
exponent
 The exponent of a number says
how many times to use the
number in a multiplication.
 Exponents are also called
Powers or Indices.
 In words: 52
could be called “5
to the second power”, “5 to the
power of 2” or simply “ 5
squared”.

4. Here are the Laws (explanations follow):
Law Example
x1
= x 61
= 6
x0
= 1 70
= 1
x-1
= 1/x 4-1
= 1/4
xm
• xn
= xm+n
x2
• x3
= x2+3
= x5
xm
/xn
= xm-n
x6
/x2
= x6-2
= x4
(xm
)n
= Xmn
(x2
)3
= X2(3)
= x6
(xy)
n
= x
n
• y
n
(xy)3
= x3
y3
(x/y)
n
= x
n
/y
n
(x/y)
2
= x
2
/y
2
Laws of Exponents

5. The first three laws (x1
= x, x0
= 1 and x-1
= 1/x) are
just part of the natural sequence of exponents. Have a
look at this:
Example: Powers of 5
..etc..
5xLarger
5xSmaller
5
2
1 • 5 • 5 25
51
1 • 5 5
50
1 1
5
-1
1 ÷ 5 0.2
5
-2
1 ÷ 5 ÷ 5 0.04
..etc..
Notice that positive, zero or negative exponents are
really part of the same pattern depending on whether
exponents gets larger (or smaller).
Laws of Exponents

6. Exponential form: The exponent of a power
indicates how many times the base multiply itself.
Laws of Exponents
Xn
= X • X • X • • • X • X • X • X
n - times
Example: 53
= 5 • 5 • 5

7. Zero Law of Exponents: Any base powered by zero
exponent equals one.
Laws of Exponents
x0
= 1
Example: 1120
= 1
(flower)0
= 1

8. Negative Law of Exponents: If the base is powered
by the negative exponent, then the base becomes
reciprocal with the positive exponent.
Laws of Exponents
X-1
= 1/x
Example #1: 2-3
= 1/23
= 1/8
Example #2: 1/5-3
= 53
/1 = 53
= 125

9. Multiplicative Law of Exponents: If the bases are
the same and if the operations between the bases is
multiplication, then the result is the base powered by
the sum of individual exponents.
Laws of Exponents
xm
• xn
= xm+n
Example: 23
• 24
= 23+4
= 27
Proof: 23
• 24
= ( 2 • 2 • 2 ) • (2 • 2 • 2 •2 ) = 2 • 2 •
2 • 2 • 2 • 2 • 2 ) = 27

10. Division Law of Exponents: If the bases are the
same and if the operations between the bases is
division, then the result is the base powered by the
difference of individual exponents.
Laws of Exponents
xm
/xn
= xm-n
Example: 54
/53
= x4-3
= 51
= 5
Proof: 54
/53
= 5 • 5 • 5 • 5 / 5 • 5 • 5 = 5

11. Exponential Law of Exponents: If the exponential
form is powered by another exponent, then the result
is the base powered by the product of individual
exponents.
Laws of Exponents
(xm
)n
= Xmn
Example: (43
)2
= 43•2
= 46
Proof: (43
)2
= ( 4 • 4 • 4 )2
= ( 4 • 4 • 4 ) • ( 4 • 4 • 4 )
= 4 • 4 • 4 • 4 • 4 • 4 = 46

12. Product Law of Exponents: If the product of the
bases is powered by the same exponent, then the result
is a multiplication of individual factors of the product,
each powered by the given exponent.
Laws of Exponents
(xy)n
= xn
• yn
Example: 36 = 62
= ( 2 • 3 ) 2
= 2 2
• 3 2
Proof: 2 2
• 3 2
= 4 • 9 = 36

13. Quotient Law of Exponents: If the quotient of the
bases is powered by the same exponent, then the result
is both numerator and denominator , each powered by
the given exponent.
Laws of Exponents
(x/y)n
= xn
/yn
Example: (2/7)3
= 23
/73

14. Evaluation
Simplify:
Laws of Exponents
 43
.2 a
 3
3x2.3
 52
3.1
22
)2x3(.4 
2
5
9
3
3
.5 





 042
.6 ts
32
)3x()3(.7 
7
)42(.8 
4
12
.9
s
s
 02
3.10

15. Reference
www.mathisfun.com/algebra/exponent-laws.html