exponents and powers by arjun rastogi

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