**Exponents and Powers**

**1. Find the value of:**

**(i) 2 ^{6}**

**Solution:-**

The above value can be written as,

= 2 × 2 × 2 × 2 × 2 × 2

= 64

**(ii) 9 ^{3}**

**Solution:-**

The above value can be written as,

= 9 × 9 × 9

= 729

**(iii) 11 ^{2}**

**Solution:-**

The above value can be written as,

= 11 × 11

= 121

**(iv) 5 ^{4}**

**Solution:-**

The above value can be written as,

= 5 × 5 × 5 × 5

= 625

**2. Express the following in exponential form:**

**(i) 6 × 6 × 6 × 6**

**Solution:-**

The given question can be expressed in the exponential form as 6^{4}.

**(ii) t × t**

**Solution:-**

The given question can be expressed in the exponential form as t^{2}.

**(iii) b × b × b × b**

**Solution:-**

The given question can be expressed in the exponential form as b^{4}.

**(iv) 5 × 5× 7 × 7 × 7**

**Solution:-**

The given question can be expressed in the exponential form as 5^{2} × 7^{3}.

**(v) 2 × 2 × a × a**

**Solution:-**

The given question can be expressed in the exponential form as 2^{2} × a^{2}.

**(vi) a × a × a × c × c × c × c × d**

**Solution:-**

The given question can be expressed in the exponential form as a^{3} × c^{4} × d.

**3. Express each of the following numbers using exponential notation:**

**(i) 512**

**Solution:-**

The factors of 512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

So it can be expressed in the exponential form as 2^{9}.

**(ii) 343**

**Solution:-**

The factors of 343 = 7 × 7 × 7

So it can be expressed in the exponential form as 7^{3}.

**(iii) 729**

**Solution:-**

The factors of 729 = 3 × 3 × 3 × 3 × 3 × 3

So it can be expressed in the exponential form as 3^{6}.

**(iv) 3125**

**Solution:-**

The factors of 3125 = 5 × 5 × 5 × 5 × 5

So it can be expressed in the exponential form as 5^{5}.

**4. Identify the greater number, wherever possible, in each of the following?**

**(i) 4 ^{3} or 3^{4}**

**Solution:-**

The expansion of 4^{3} = 4 × 4 × 4 = 64

The expansion of 3^{4} = 3 × 3 × 3 × 3 = 81

Clearly,

64 < 81

So, 4^{3} < 3^{4}

Hence 3^{4} is the greater number.

**(ii) 5 ^{3} or 3^{5}**

**Solution:-**

The expansion of 5^{3} = 5 × 5 × 5 = 125

The expansion of 3^{5} = 3 × 3 × 3 × 3 × 3= 243

Clearly,

125 < 243

So, 5^{3} < 3^{5}

Hence 3^{5} is the greater number.

**(iii) 2 ^{8} or 8^{2}**

**Solution:-**

The expansion of 2^{8} = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256

The expansion of 8^{2} = 8 × 8= 64

Clearly,

256 > 64

So, 2^{8} > 8^{2}

Hence 2^{8} is the greater number.

**(iv) 100 ^{2} or 2^{100}**

**Solution:-**

The expansion of 100^{2} = 100 × 100 = 10000

The expansion of 2^{100}

2^{10} = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024

Then,

2^{100 }= 1024 × 1024 ×1024 × 1024 ×1024 × 1024 × 1024 × 1024 × 1024 × 1024 =

Clearly,

100^{2} < 2^{100}

Hence 2^{100} is the greater number.

**(v) 2 ^{10} or 10^{2}**

**Solution:-**

The expansion of 2^{10} = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024

The expansion of 10^{2} = 10 × 10= 100

Clearly,

1024 > 100

So, 2^{10} > 10^{2}

Hence 2^{10} is the greater number.

**5. Express each of the following as product of powers of their prime factors:**

**(i) 648**

**Solution:-**

Factors of 648 = 2 × 2 × 2 × 3 × 3 × 3 × 3

= 2^{3 }× 3^{4}

**(ii) 405**

**Solution:-**

Factors of 405 = 3 × 3 × 3 × 3 × 5

= 3^{4} × 5

**(iii) 540**

**Solution:-**

Factors of 540 = 2 × 2 × 3 × 3 × 3 × 5

= 2^{2 }× 3^{3} × 5

**(iv) 3,600**

**Solution:-**

Factors of 3600 = 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5

= 2^{4 }× 3^{2} × 5^{2}

**6. Simplify:**

**(i) 2 × 10 ^{3}**

**Solution:-**

The above question can be written as,

= 2 × 10 × 10 × 10

= 2 × 1000

= 2000

**(ii) 7 ^{2} × 2^{2}**

**Solution:-**

The above question can be written as,

= 7 × 7 × 2 × 2

= 49 × 4

= 196

**(iii) 2 ^{3} × 5**

**Solution:-**

The above question can be written as,

= 2 × 2 × 2 × 5

= 8 × 5

= 40

**(iv) 3 × 4 ^{4}**

**Solution:-**

The above question can be written as,

= 3 × 4 × 4 × 4 × 4

= 3 × 256

= 768

**(v) 0 × 10 ^{2}**

**Solution:-**

The above question can be written as,

= 0 × 10 × 10

= 0 × 100

= 0

**(vi) 5 ^{2} × 3^{3}**

**Solution:-**

The above question can be written as,

= 5 × 5 × 3 × 3 × 3

= 25 × 27

= 675

**(vii) 2 ^{4} × 3^{2}**

**Solution:-**

The above question can be written as,

= 2 × 2 × 2 × 2 × 3 × 3

= 16 × 9

= 144

**(viii) 3 ^{2} × 10^{4}**

**Solution:-**

The above question can be written as,

= 3 × 3 × 10 × 10 × 10 × 10

= 9 × 10000

= 90000

**7. Simplify:**

**(i) (– 4) ^{3}**

**Solution:-**

The expansion of -4^{3}

= – 4 × – 4 × – 4

= – 64

**(ii) (–3) × (–2) ^{3}**

**Solution:-**

The expansion of (-3) × (-2)^{3}

= – 3 × – 2 × – 2 × – 2

= – 3 × – 8

= 24

**(iii) (–3) ^{2} × (–5)^{2}**

**Solution:-**

The expansion of (-3)^{2} × (-5)^{2}

= – 3 × – 3 × – 5 × – 5

= 9 × 25

= 225

**(iv) (–2) ^{3} × (–10)^{3}**

**Solution:-**

The expansion of (-2)^{3} × (-10)^{3}

= – 2 × – 2 × – 2 × – 10 × – 10 × – 10

= – 8 × – 1000

= 8000

**8. Compare the following numbers:**

**(i) 2.7 × 10 ^{12} ; 1.5 × 10^{8}**

**Solution:-**

By observing the question

Comparing the exponents of base 10,

Clearly,

2.7 × 10^{12} > 1.5 × 10^{8}

**(ii) 4 × 10 ^{14} ; 3 × 10^{17}**

**Solution:-**

By observing the question

Comparing the exponents of base 10,

Clearly,

4 × 10^{14} < 3 × 10^{17}