Laws of Exponents



The laws of exponents are explained here along with their examples.

1. Multiplying powers with same base

For example: x2 × x3, 23 × 25, (-3)2 × (-3)4

In multiplication of exponents if the bases are same then we need to add the exponents.



Consider the following:

1. 23 × 22= (2 × 2 × 2) × (2 × 2) = 23 + 2 = 25

2. 34 × 32 = (3 × 3 × 3 × 3) × (3 × 3) = 34 + 2 = 36

3. (-3)3 × (-3)4 = [(-3) × (-3) × (-3)] × [(-3) × (-3) × (-3) × (-3)]

= (-3)3 + 4 = (-3)7

4. m5 × m3 = (m × m × m × m × m) × (m × m × m) = m5 + 3 = m8

From the above examples, we can generalize that during multiplication when the bases are same then the exponents are added.

am × an = am+n

In other words, if ‘a’ is a non-zero integer or a non-zero rational number and m and n are positive integers, then

am × an = am + n

Similarly, (a/b)m × (a/b)n = (a/b)m + n

Note:

(i) Exponents can be added only when the bases are same.

(ii) Exponents cannot be added if the bases are not same like

m5 × n7, 23 × 34

For example:

1. 53 ×56

= (5 × 5 × 5) × (5 × 5 × 5 × 5 × 5 × 5)

= 53+6[here the exponents are added]

= 59


2. (-7)10 × (-7)12

= [(-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7)] × [( -7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7)].

= (-7)10+12 [exponents are added]

= (-7)22


3. (1/2)4 × ( 1/2)3

=[(1/2) × ( 1/2) × ( 1/2) × ( 1/2)] × [ ( 1/2) × ( 1/2) × ( 1/2)]

=(1/2)4 + 3

=(1/2)7

4. 32 × 35

= 32 + 5

= 37


5. (-2)7 × (-2)3

= (-2)7 + 3

= (-2)10


6. (4/9)3 × (4/9)2

= (4/9)3 + 2

= (4/9)5

We observe that the two numbers with the same base are

multiplied; the product is obtained by adding the exponent.



2. Dividing powers with the same base

For example:

35 ÷ 31, 22 ÷ 21, 5(2) ÷ 53

In division if the bases are same then we need to subtract the exponents.

Consider the following:

27 ÷ 24 = 27/24 = (2 × 2 × 2 × 2 × 2 × 2 × 2)/(2 × 2 × 2 × 2) = 27 - 4

= 23

56 ÷ 52 = 56/52 = (5 × 5 × 5 × 5 × 5 × 5)/(5 × 5) = 56-2 = 54

105 ÷ 103 = 105/103 = (10 × 10 × 10 × 10 × 10)/(10 × 10 × 10) = 105 - 3

= 102

74 ÷ 75 = 74/75 = (7 × 7 × 7 × 7)/(7 × 7 × 7 × 7 × 7) = 74 - 5 = 7-1

Let a be a non zero number, then

a5 ÷ a3 = a5/a3 = (a × a × a × a × a)/(a × a × a) = a5 - 3 = a2

again, a3 ÷ a5 = a3/a5 = (a × a × a)/(a × a × a × a × a) = a-(5 - 3 ) = a-2

Thus, in general, for any non-zero integer a,

am ÷ an = am/an = am - n

Note 1:

Where m and n are whole numbers and m > n;

am ÷ an = am/an = a-(n - m)

Note 2:

Where m and n are whole numbers and m < n;

We can generalize that if ‘a’ is a non-zero integer or a non-zero rational number and m and n are positive integers, such that m > n, then

am ÷ an = am - n if m < n, then am ÷ an = 1/an - m

Similarly, (a/b)m ÷ (a/b)n = (a/b)m -n

For example:

1. 710 ÷ 78 = 710/78

= (7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7)/(7 × 7 × 7 × 7 × 7 × 7 × 7 × 7)

= 710 - 8 [here exponents are subtracted]

= 72


2. p6 ÷ p1=p6/p1

= (p × p × p × p × p × p)/p

= p6 - 1 [here exponents are subtracted]

= p5


3. 44 ÷ 42 = 44/42

= (4 × 4 × 4 × 4)/(4 × 4)

= 44 - 2 [here exponents are subtracted]

= 42


4. 102 ÷ 104 = 102/104

= (10 × 10)/(10 × 10 × 10 × 10)

= 10-(4 - 2) [See note (2)]

= 10-2


5. 53 ÷ 51

= 53 - 1

= 52


6. 35/32

= 35 - 2

= 33


7. (-5)9/(-5)6

= (-5)9 - 6

= (-5)3


8. (7/2)8 ÷ (7/2)5

= (7/2)8 - 5

= (7/2)3



3. Power of a power

For example: (23)2, (52)6, (32 )-3

In power of a power you need multiply the powers.

Consider the following

(i) (23)4

Now, (23)4 means 23 is multiplied four times

i.e. (23)4 = 23 × 23 × 23 × 23

=23 + 3 + 3 + 3

=212

Note: by law (l), since am × an = am + n.


(ii) (23)2

Similarly, now (23)2 means 23 is multiplied two times

i.e. (23)2 = 23 × 23

= 23 + 3 [since am × an = am + n]

= 26

Note: Here, we see that 6 is the product of 3 and 2 i.e,(23)2 = 23 × 2 = 26


(iii)(4-2)3

Similarly, now (4-2 )3 means 4-2 is multiplied three times

i.e. (4-2)3 =4-2 × 4-2 × 4-2

=4-2 + (-2) + (-2)

=4-2 -2 -2

=4-6

Note: Here, we see that -6 is the product of -2 and 3 i.e, (4-2)3

= 4-2 × 3 =4-6


For example:

1.(32)4 = 32 × 4 = 38

2. (53)6 = 53 × 6 = 518

3. (43)8 = 43 × 8 = 424

4. (am)4 = am × 4 = a4m

5. (23)6 = 23 × 6 = 218

6. (xm)-n = xm × (-n) = x-mn

7. (52)7 = 52 × 7 = 514

8. [(-3)4]2 = (-3)4 × 2 = (-3)8

In general, for any non-integer a, (am)n=am × n =amn.Thus where m and n are whole numbers.

If ‘a’ is a non-zero rational number and m and n are positive integers, then {(a/b)m}n = (a/b) mn

For example:

[(-2/5)3]2

= (-2/5)3 × 2

= (-2/5)6



4. Multiplying powers with the same exponents

For example: 32 × 22, 53 × 73

We consider the product of 42 and 32, which have different bases, but the same exponents.

(i) 42 × 32 [here the powers are same and the bases are different]

= (4 × 4)×(3 × 3)

= (4 × 3)×(4 × 3)

= 12 × 12

= 122

Here, we observe that in 122, the base is the product of bases 4 and 3.

We consider,

(ii) 43 × 23

=(4 × 4 × 4)×(2 × 2 × 2)

=(4 × 2)× ( 4 × 2)× (4 × 2)

=8 × 8 × 8

=83

(iii) We also have, 23 × a3

= (2× 2 × 2)×(a × a × a)

= (2 × a)×(2 × a)×(2 × a)

= (2 × a)3

= (2a)3 [here 2 × a = 2a]


(iv) Similarly, we have, a3 × b3

= (a × a × a)×(b × b × b)

= (a × b)× (a × b)× (a × b)

= (a × b)3

= (ab)3 [here a × b = ab]

Note: In general, for any non-zero integer a, b.

am × bm

= (a × b)m

= (ab)m [here a × b = ab]

Note: Where m is any whole number.

(-a)3 × (-b)3

= [(-a) × (-a) × (-a)]×[(-b) × (-b) × (-b)]

= [(-a) × (-b)]× [(-a) × (-b)]× [(-a) × (-b)]

= [(-a)×(-b)]3

= (ab)3 [here a × b = ab and two negative become positive,

(-) × (-) = +]



5. Negative Exponents

If the exponent is negative we need to change it into positive exponent by writing the same in the denominator and 1 in the numerator.

If ‘a’ is a non-zero integer or a non-zero rational number and m is a positive integers, thena-m is the reciprocal of am, i.e.,

a-m = 1/am, if we take ‘a’ as p/q then (p/q)-m = 1/(p/q)m = (q/p)m

again, 1/a-m = am

Similarly, (a/b)-n = (b/a)n, where n is a positive integer

Consider the following

2-1 = 1/2

2-2 = 1/22 = 1/2 × 1/2 = 1/4

2-3 = 1/23 = 1/2 × 1/2 × 1/2 = 1/8

2-4 = 1/24 = 1/2 × 1/2 × 1/2 × 1/2 = 1/16

2-5 = 1/25 = 1/2 × 1/2 × 1/2 × 1/2 × 1/2 = 1/32

[So in negative exponent we need to write 1 in the numerator and in the denominator 2 multiplied to itself five times as 2-5. In other words negative exponent is the reciprocal of positive exponent]

For example:

1. 10-3

= 1/10-3 [here we can see that 1 is in the numerator and in the denominator 103 as we know that negative exponent is the reciprocal]

= 1/10 × 1/10 × 1/10 [here 10 is multiplied to itself 3 times]

= 1/1000


2. (-2)-4

= 1/(-2)4 [here we can see that 1 is in the numerator and in the denominator (-2)4]

=(- 1/2) × (- 1/2) × (- 1/2) × (- 1/2)

= 1/16


3. 2-5

= 1/25

= 1/2 × 1/2

= 1/4


4. 1/3-4

= 34

= 3 × 3 × 3 × 3

= 81


5. (-7)-3

= 1/(-7)3


6. (3/5-3

= (5/3)3


7. (-7/2)-2

= (-2/7)2



6. Power with exponent zero

If the exponent is 0 then you get the result 1 whatever the base is.

For example: 80, ( a/b)0, m0….

If ‘a’ is a non-zero integer or a non-zero rational number then,

a0 = 1

Similarly, (a/b)0 = 1

Consider the following

a0 = 1[anything to the power 0 is 1]

(a/b)0 = 1

(-2/3)0 = 1

(-3)0 = 1

For example:

1. (2/3)3 × (2/3)-3

= (2/3)3 + (-3) [here we know that am × an = am+ n]

= (2/3)3 - 3

= (2/3)0

= 1


2. 25 ÷ 25

= 25/25

= (2 × 2 × 2 × 2 × 2)/(2 × 2 × 2 × 2 × 2)

= 25 - 5 [here by the law am ÷ an =am - n]

= 20

= 1


3. 40 × 30

= 1 × 1[here as we know anything to the power 0 is 1]

= 1


4. am × a-m

= am - m

= a0

= 1


5. 50 = 1

6. (-4/9)0 = 1

7. (-41)0 = 1

8. (3/7)0 = 1



7. Fractional Exponent

In fractional exponent we observe that the exponent is in fraction form.

Consider the following:

21/1 = 2 (it will remain 2).

21/2 =√2 (square root of 2).

21/3 =∛2 (cube root of 2).

21/4 =∜2 (fourth root of 2).

21/5 =5√2 (fifth root of 2).

For example:

1. a1/n [Here a is called the base and 1/n is called the exponent or power]

= n√a [nth root of a]

2. 31/2= √3 [square root of 3]

3. 51/3 = ∛5 [cube root of 5]



Exponents

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  • Integral Exponents of a Rational Numbers
  • Solved Examples on Exponents
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