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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, (ab)ᵐ × (ab)ⁿ = (ab)m+n

(ab)m×(ab)n=(ab)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

Multiplying Powers with same Base, Laws of Exponents

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. (12)4 × (12)3

=[(12) × (12) × (12) × (12)] × [(12) × (12) × (12)] 


=(12)4+3

=(12)⁷


4. 3² × 3⁵

= 32+5

= 3⁷


5. (-2)⁷ × (-2)³

= (-2)7+3

= (-2)10



6. (49)³ × (49

= (49)3+2

= (49)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 = 2724

            = 2×2×2×2×2×2×22×2×2×2

            = 274

            = 2³

5⁶ ÷ 5² = 5652

            = = 5×5×5×5×5×55×5

            = 562 

            = 5⁴


10⁵ ÷ 10³ = 105103

                = 10×10×10×10×1010×10×10

                = 1053

                = 10²


7⁴ ÷ 7⁵ = 7475

            = 7×7×7×77×7×7×7×7

            = 745 

            = 71


Let a be a non zero number, then

a⁵ ÷ a³ = a5a3

            = a×a×a×a×aa×a×a

            = a53 

            = a²


again, a³ ÷ a⁵ = a3a5

                     = a×a×aa×a×a×a×a

                     = a(53)

                     = a2

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

aᵐ ÷ aⁿ = aman = amn


Note 1: 

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

aᵐ ÷ aⁿ = aman = a(nm)


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 = amn if m < n, then aᵐ ÷ aⁿ = 1anm

Similarly, (ab)m ÷ (ab)n = ab mn

Dividing Powers with the same Base, Laws of Exponents

For example:

1. 710 ÷ 7⁸ = 71078

                             = 7×7×7×7×7×7×7×7×7×77×7×7×7×7×7×7×7

                             = 7108, [here exponents are subtracted] 

                             = 7²


2. p⁶ ÷ p¹ = p6p1

               = p×p×p×p×p×pp

               = p61, [here exponents are subtracted] 

               = p⁵


3. 4⁴ ÷ 4² = 4442

                = 4×4×4×44×4

                = 442, [here exponents are subtracted] 

                = 4²


4. 10² ÷ 10⁴ = 102104

                   = 10×1010×10×10×10

                   = 10(42)[See note (2)] 

                   = 102


5. 5³ ÷ 5¹

= 531

= 5²



6. (3)5(3)2

= 352

= 3³


7. (5)9(5)6

= (-5)96

= (-5)³


8. (72)⁸ ÷ (72)⁵

= (72)85

= (72


Laws of Exponents or Indices

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) (42


Similarly, now (42)³ means 42

 is multiplied three times


i.e. (42)³ =42 × 42 × 42

= 42+(2)+(2)

= 4222

= 46

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

                (42)³ = 42×3 = 46


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) = xmn

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 {(ab)ᵐ}ⁿ = (ab)mn

Power of a Power, Laws of Exponents

For example:

[(25)³]²

= (25)3×2

= (25)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. 

Multiplying Powers with the same Exponents, Exponent Rules

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]


aᵐ × bᵐ = (ab)ᵐ


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, then am is the reciprocal of aᵐ, i.e., 


am = 1am, if we take ‘a’ as pq then (pq)m = 1(pq)m = (qp)ᵐ


again, 1am = aᵐ


Similarly, (ab)n = (ba)ⁿ, where n is a positive integer


Consider the following

21 = 12

22 = 122 = 12 × 12 = 14

23 = 123 = 12 × 12 × 12 = 18

24 = 124 = 12 × 12 × 12 × 12  = 116

25 = 125 = 12 × 12 × 12 × 12 × 12132


[So in negative exponent we need to write 1 in the numerator and in the denominator 2 multiplied to itself five times as 25. In other words negative exponent is the reciprocal of positive exponent] 

Negative Exponents, Laws of Exponents

For example:

1. 103

= 1103, [here we can see that 1 is in the numerator and in the denominator 10³ as we know that negative exponent is the reciprocal] 

= 110 × 110 × 110, [Here 10 is multiplied to itself 3 times] 

= 11000



2. (-2)4

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

= (- 12) × (- 12) × (- 12) × (- 12

= 116


3. 25

= 125

= 12 × 12

= 14



4. 134

= 3⁴

= 3 × 3 × 3 × 3

= 81


5. (-7)3

1(7)3



6. (35)3

= (53



7. (-72)2

= (-27


6. Power with Exponent Zero

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

For example: 80, (ab)0, m0…....


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

a0 = 1


Similarly, (ab)0 = 1


Consider the following

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

(ab)0 = 1

(23)0 = 1

(-3)0 = 1

Power with Exponent Zero, Laws of Exponents

For example:

1. (23)³ × (23)3

= (23)3+(3), [Here we know that aᵐ × aⁿ = am+n

= (23)33

= (23)0

= 1



2. 2⁵ ÷ 2⁵

2525

= 2×2×2×2×22×2×2×2×2

= 255, [Here by the law aᵐ ÷ aⁿ =amn

= 2

= 1




3. 40 × 30

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

= 1


4. aᵐ × am

= amm

= a0

= 1


5. 50 = 1

6. (49)0 = 1

7. (-41)0 = 1

8. (37)0 = 1



7. Fractional Exponent

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

a1n, [Here a is called the base and 1n is called the exponent or power]

na, [nth root of a] 

a1n=na



Consider the following:

211 = 2 (it will remain 2). 

212 = √2 (square root of 2).

213 = ∛2 (cube root of 2).

214 = ∜2 (fourth root of 2).

215 = 52 (fifth root of 2). 

Fractional Exponent, Laws of Exponents

For example:

1. 212 = √2 (square root of 2). 

2. 312 = √3 [square root of 3] 

3. 513 = ∛5 [cube root of 5]

4. 1013 = ∛10 [cube root of 10]

5. 2117 = 721 [Seventh root of 21]


 Exponents

Exponents

Laws of Exponents

Rational Exponent

Integral Exponents of a Rational Numbers

Solved Examples on Exponents

Practice Test on Exponents


 Exponents - Worksheets

Worksheet on Exponents












8th Grade Math Practice

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