grade 7 exponents and powersrule 1: when we multiply two numbers with the same base, we add the...

Grade 7Exponents and Powers

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Answer the questions

(1) Solve the following and write in the simplest fraction form.

(2)

(3)

(4) Find the value of the following:

A) 82 B) (-9)2

C) 62 D) (-10)3

(5) If 2p + 2p + 1 = 12, find the value of p.

(6) Write the number for the following expanded forms:

A) 5×101 + 0×106 + 4×102 + 5×102 + 0×101 + 3×108 + 9×100

B) 0×105 + 0×100 + 6×101 + 6×108 + 0×104

(7) Simplify the following and write the answer in the exponential form

A) 78 × 74

73 × 76 × 77 × 78 × 79 × 79 B)

118 × 115

119 × 114 × 116 × 117

C) 54 × 59

59 × 59 × 52 × 55 D)

173 × 173 × 178

175 × 177 × 177 × 172 × 177 × 172

(8) If x = 1 and y = 5, find the value of .

(9) = ?

(10)

Find the value of x.

(11) Solve the following and write in the simplest form of fraction.

ID : ae-7-Exponents-and-Powers [1]

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(12) Ferran plants a dandelion on his 8 th birthday. If the plant has one dandelion to start with and thenumber of dandelions doubles every week then how many dandelions will be there after x weeks?

(13) Simplify the following and write the answer in the exponential form.

A) 32 × 324

98 × 32 × 278 × 99 B)

34 × 36 × 33 × 34 × 35 × 36 × 276

32 × 273 × 99

C) 89

49 × 29 D)

96 × 311

279 × 93

(14)

Find the value of x.

(15) = ?

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Answers

(1) 4

29

Step 1

{( 2

3 )0 + (

2

5 )-2}-1 can be simplified as:

1

( 2

3 )0 + (

2

5 )-2

=

1

( 2

3 )0 +

1

( 2

5 )2

=

1

(2)0

(3)0 +

1

(2)2

(5)2

=

1

(2)0

(3)0 +

(5)2

(2)2

=

1

1

1 +

25

4

=

1

29

4

= 4

29

Step 2

Hence, the simplest fraction form of {( 2

3 )0 + (

2

5 )-2}-1 is

4

29 .



(2) 125

36

Step 1

Here, we have to find the value of ( 5

4 )3 × (

-4

3 )2 .

Step 2

Let us solve each term separately:

( 5

4 )3 =

125

64

and

( -4

3 )2 =

16

9

Step 3

Now, ( 5

4 )3 × (

-4

3 )2 =

125

64 ×

16

9

= 125

36

Step 4

Therefore, the value of ( 5

4 )3 × (

-4

3 )2 is

125

36 .



(3) -3

8

Step 1

Here, we have to find out the value of:

( -3

2 )3 × (

2

3 )2 × (

1

2 )2

Step 2

Let us solve each term separately:

( -3

2 )3 =

-27

8 ,

( 2

3 )2 =

4

9

and

( 1

2 )2 =

1

4

Step 3

Now, ( -3

2 )3 × (

2

3 )2 × (

1

2 )2 can also be written as:

= -27

8 ×

4

9 ×

1

4

= -3

8

(4) A) 64

Step 1

82 can be written as:

82 = 8 × 8 = 64

Step 2

Hence, the value of 82 is 64.



B) 81

Step 1

(-9)2 can be written as:

(-9)2 = (-9) × (-9) = 81

Step 2

Hence, the value of (-9)2 is 81.

C) 36

Step 1

62 can be written as:

62 = 6 × 6 = 36

Step 2

Hence, the value of 62 is 36.

D) -1000

Step 1

(-10)3 can be written as:

(-10)3 = (-10) × (-10) × (-10) = -1000

Step 2

Hence, the value of (-10)3 is -1000.



(5) 2

Step 1

2p + 2p + 1 = 12 can be solved for the value of p, by the help of the following steps:

2p + 2p + 1 = 12

⇒ 2p + (2p 21) = 12

⇒ 2p(1 + 21) = 12

⇒ 2p(1 + 2) = 12

⇒ 2p = 12

3

⇒ 2p = 22

Step 2

The above condition is satisfied only when the value of p is 2. Hence, the value of p is 2.

(6) A) 300000959

Step 1

5×101 + 0×106 + 4×102 + 5×102 + 0×101 + 3×108 + 9×100 can be solved by thefollowing steps:

5×101 + 0×106 + 4×102 + 5×102 + 0×101 + 3×108 + 9×100

= 50 + 0 + 400 + 500 + 0 + 300000000 + 9= 300000959

Step 2

Hence, the number for the expended form 5×101 + 0×106 + 4×102 + 5×102 + 0×101 +

3×108 + 9×100 is 300000959.

B) 600000060

Step 1

0×105 + 0×100 + 6×101 + 6×108 + 0×104 can be solved by the following steps:

0×105 + 0×100 + 6×101 + 6×108 + 0×104

= 0 + 0 + 60 + 600000000 + 0= 600000060

Step 2

Hence, the number for the expended form 0×105 + 0×100 + 6×101 + 6×108 + 0×104 is600000060.



(7) A) 736

Step 1

Rules of the exponent:Rule 1: When we multiply two numbers with the same base, we add the exponents.Rule 2: When we divide two numbers with the same base, we subtract the exponents.Rule 3: When we have an exponent expression that is raised to a power, we canmultiply the exponent and the power.

Step 2

78 × 74

73 × 76 × 77 × 78 × 79 × 79 can be simplified as:

78 × 74

73 × 76 × 77 × 78 × 79 × 79

= 7(8 + 4)

7(3 + 6) × 7(7 + 8 + 9 + 9)

= 712

79 × 733

= 712 ÷ 79 × 733

= 7(12 - 9 + 33)

= 736

Step 3

Hence, the exponential form of 78 × 74

73 × 76 × 77 × 78 × 79 × 79 is 736.



B) 1121

Step 1


Step 2

118 × 115

119 × 114 × 116 × 117 can be simplified as:

118 × 115

119 × 114 × 116 × 117

= 11(8 + 5)

11(9) × 11(4 + 6 + 7)

= 1113

119 × 1117

= 1113 ÷ 119 × 1117

= 11(13 - 9 + 17)

= 1121

Step 3


119 × 114 × 116 × 117 is 1121.



C) 52

Step 1


Step 2

54 × 59


54 × 59

59 × 59 × 52 × 55

= 5(4 - 9)

5(9 + 9) × 5(2 + 5)

= 54

518 × 57

= 54 ÷ 518 × 57

= 5(4 - 18 + 7)

= 52

Step 3


59 × 59 × 52 × 55 is 52.



D) 172

Step 1


Step 2

173 × 173 × 178

175 × 177 × 177 × 172 × 177 × 172 can be simplified as:

173 × 173 × 178

175 × 177 × 177 × 172 × 177 × 172

= 17(3 + 3 - 8)

17(5 + 7 + 7 + 2) × 17(7 + 2)

= 176

1721 × 179

= 176 ÷ 1721 × 179

= 17(6 - 21 + 9)

= 172

Step 3

Hence, the exponential form of 173 × 173 × 178

175 × 177 × 177 × 172 × 177 × 172 is 172.



(8) 1

5

Step 1

According to the question, we have to find out the value of ( x

y )x, where x = 1 and y = 5.

Step 2

Let us put the values of x and y in ( x

y )x

( x

y )x = (

1

5 )1

= 1

5 .

Step 3

Hence, the value of ( x

y )x, where x = 1 and y = 5 is

1

5 .



(9) 1

9

Step 1

Rules of the exponent:Rule 1: When we multiply any two numbers with the same base, we add the exponents.Rule 2: When we divide any two numbers with the same base, we subtract the exponents.Rule 3: When we have an exponent expression that is raised to a power, we can multiply theexponent and the power.

Rule 4: (x)-n can be written as 1

(x)n

Step 2

Now, [ { (- 1

3 )1}-1]-2 can be simplified as:

= { (- 1

3 )1} -1 × -2

={ (- 1

3 )1}2

= (- 1

3 ) 1 × 2

=(- 1

3 )2

= 1

9

Step 3

Hence, the value of [ { (- 1

3 )1}-1]-2 is

1

9 .



(10) -1

Step 1



(x)n

Rule 5: If, (y)x = (y)10 then we can write x = 10.

Step 2

Now

{( 1

3 )-1}x =

1

3

⇒ ( 1

3 )-1 × x =

1

3

⇒ ( 1

3 )(-1)(x) = (

1

31 )

⇒ ( 1

3 )(-1)(x) = (

1

3 )1

Comparing both sides: ⇒ (-1)(x) = 1 ⇒ x = -1

Step 3

Hence, the value of x is -1 .



(11) 65

1

Step 1

can be simplified as:

+

= (3)3

(3)3 +

(4)3

(1)3

= 27

27 +

64

1

= 65

1

Step 2

Hence, the simplest fraction form of is 65

1 .

(12) 2x

Step 1

According to the question, Ferran plants a dandelion on his 8 th birthday. The plant has onedandelion in the beginning and the number of dandelions doubles every week.

Step 2

Number of dandelion in the beginning = 1

Number of dandelion after 1 week = 21

Number of dandelion after 2 weeks = 22

Number of dandelion after 3 weeks = 23

--------------------------------- ---------------------------------

Number of dandelion after x weeks = 2x

Step 3

Hence, the total number of dandelions after x weeks will be 2x.



(13) A) 32

Step 1


Step 2

32 × 324


32 × 324

98 × 32 × 278 × 99

= (3)2 × 324

(3 × 3)8 × (3)2 × (3 × 3 × 3)8 × (3 × 3)9

= (3)2 × 324

(32)8 × (3)2 × (33)8 × (32)9

=

3(2 + 24)

3(16 + 2 +

24)

× 3(18)

= 326

342 × 318

= 326 ÷ 342 × 318

= 3(26 - 42 + 18)

= 32

Step 3


98 × 32 × 278 × 99 is 32.



B) 353

Step 1


Step 2

34 × 36 × 33 × 34 × 35 × 36 × 276

32 × 273 × 99 can be simplified as:

34 × 36 × 33 × 34 × 35 × 36 × 276

32 × 273 × 99

= (3)4 × (3)6 × (3)3 × (3)4 × (3)5 × (3)6 × (3 × 3 × 3)6

(3)2 × (3 × 3 × 3)3 × (3 × 3)9

= (3)4 × (3)6 × (3)3 × (3)4 × (3)5 × (3)6 × (33)6

(3)2 × (33)3 × (32)9

=

3(4 + 6 + 3 + 4 + 5 + 6 +

18)

3(2 + 9)

× 3(18)

= 346

311 × 318

= 346 ÷ 311 × 318

= 3(46 - 11 + 18)

= 353

Step 3

Hence, the exponential form of 34 × 36 × 33 × 34 × 35 × 36 × 276

32 × 273 × 99 is 353.



C) 218

Step 1


Step 2

89

49 × 29 can be simplified as:

89

49 × 29

= (2 × 2 × 2)9

(2 × 2)9 × (2)9

= (23)9

(22)9 × (2)9

= 2(27)

2(18) × 2(9)

= 227

218 × 29

= 227 ÷ 218 × 29

= 2(27 - 18 + 9)

= 218

Step 3

Hence, the exponential form of 89

49 × 29 is 218.



D) 32

Step 1


Step 2

96 × 311

279 × 93 can be simplified as:

96 × 311

279 × 93

= (3 × 3)6 × 311

(3 × 3 × 3)9 × (3 × 3)3

= (32)6 × 311

(33)9 × (32)3

=

3(12 +

11)

3(27)

× 3(6)

= 323

327 × 36

= 323 ÷ 327 × 36

= 3(23 - 27 + 6)

= 32

Step 3


279 × 93 is 32.



(14) 1

Step 1



(x)n

Rule 5: If, (y)x = (y)10 then we can write x = 10.

Step 2

Now

[{( 1

2 )1}1]x =

1

2

⇒ {( 1

2 )1}1x =

1

(2)1

⇒ ( 1

2 )1x =

1

(2)1

⇒ ( 1

2 )1x = (

1

2 )1

⇒ 1x = 1⇒ x = 1

Step 3

Hence, the value of x is 1 .



(15) 27

1000

Step 1

Remember the following rules of exponents:Rule 1: When we multiply any two numbers with the same base, we add the exponents.Rule 2: When we divide any two numbers with the same base, we subtract the exponents.Rule 3: When we have an exponent expression that is raised to a power, we can multiply theexponent and the power.


(x)n

Step 2

Now can be simplified as:

=

1

=

1

(5)3

(3)3 ÷

(2)3

(4)3

=

1

125

27 ÷

8

64

=

1

125

27 ×

64

8

=

1

8000

216

= 27

1000

Step 3

Hence, the value of is 27

1000 .



grade 7 exponents and powersrule 1: when we multiply two numbers with the same base, we add the...

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