exponents and powers by arjun rastogi
TRANSCRIPT
EXPONENTS
• A quantity representing the power to which
a given number or expression is to be
raised, usually expressed as a raised
symbol beside the number or expression
(e.g. 3 in 23 = 2 × 2 × 2).
• Exponents are shorthand for repeated multiplication of the same thing by itself. For instance, the shorthand for multiplying three copies of the number 5 is shown on the right-hand side of the "equals" sign in (5)(5)(5) =53. The "exponent", being 3 in this example, stands for however many times the value is being multiplied. The thing that's being multiplied, being5 in this example, is called the "base".
General Enquiry
Exponents
35Power
base
exponent
3 3 means that is the exponential
form of t
Example:
he number
125 5 5
.125
53 means 3 factors of 5 or 5 x 5 x 5
The Laws of Exponent
Comes From 3 ideas
• The exponent says how many times to
use the number in a multiplication.
• A negative exponent means divide,
because the opposite of multiplying is
dividing
• A fractional exponent like 1/n means
to take the nth root:
Laws Of Exponent• x1 = x
• x0 = 1
• x-1 = 1/x
• xmxn = xm+n
• xm/xn = xm-n
• (xm)n = xmn
• (xy)n = xnyn
• (x/y)n = xn/yn
• x-n = 1/xn
The Laws of Exponents:
#1: Exponential form: The exponent of a power indicates
how many times the base multiplies itself.
n
n times
x x x x x x x x
3Example: 5 5 5 5
n factors of x
#2: Multiplying Powers: If you are multiplying Powers
with the same base, KEEP the BASE & ADD the EXPONENTS!
m n m nx x x
So, I get it! When you multiply Powers, you add the exponents!
512
2222 93636
#3: Dividing Powers: When dividing Powers with the
same base, KEEP the BASE & SUBTRACT the EXPONENTS!
mm n m n
n
xx x x
x
So, I get it!
When you divide Powers, you subtract the exponents!
16
222
2 426
2
6
#4: Power of a Power: If you are raising a Power to an
exponent, you multiply the exponents!
n
m mnx x
So, when I take a Power to a power, I multiply the exponents
52323 55)5(
#5: 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.
n n nxy x y
So, when I take a Power of a Product, I apply the exponent to all factors of the product.
222)( baab
#6: 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.
n n
n
x x
y y
So, when I take a Power of a Quotient, I apply the exponent to all parts of the quotient.
81
16
3
2
3
24
44
Try these:
523.1
43.2 a
322.3 a
23522.4 ba
22 )3(.5 a
342.6 ts
5
.7t
s
2
5
9
3
3.8
2
4
8
.9rt
st
2
54
85
4
36.10
ba
ba
523.1
43.2 a
322.3 a
23522.4 ba
22 )3(.5 a
342.6 ts
SOLUTIONS
10312a
6323 82 aa
6106104232522 1622 bababa
422293 aa
1263432 tsts
5
.7t
s
2
5
9
3
3.8
2
4
8
.9rt
st
2
54
85
4
3610
ba
ba
SOLUTIONS
62232223 8199 babaab
2
822
4
r
ts
r
st
824 33
5
5
t
s
#7: Negative Law of Exponents: If the base is powered
by the negative exponent, then the base becomes reciprocal with the
positive exponent.
1m
mx
x
So, when I have a Negative Exponent, I switch the base to its reciprocal with a Positive Exponent.
Ha Ha!
If the base with the negative exponent is in the denominator, it moves to the numerator to lose its negative sign!
933
1
125
1
5
15
2
2
3
3
and
#8: Zero Law of Exponents: Any base powered by zero
exponent equals one.
0 1x
1)5(
1
15
0
0
0
a
and
a
and
So zero factors of a base equals 1. That makes sense! Every power has a coefficient of 1.
Try these:
022.1 ba
42.2 yy
15.3 a
72 4.4 ss
4323.5 yx
042.6 ts
SOLUTIONS
022.1 ba
15.3 a
72 4.4 ss
4323.5 yx
042.6 ts
1
5
1
a54s
12
81284
813
y
xyx
1
Presented by
Arjun Rastogi