**Exercise 12.1 :** Solutions of Questions on Page Number :** 197**

**Q1 : Evaluate**

**(i) 3 ^{– 2} (ii) ( – 4)^{ – 2} (iii)**

**Answer :**

(i)

(ii)

(iii)

**Q2 :Simplify and express the result in power notation with positive exponent.**

**(i) (ii)**

**(iii) (iv)**

**(v)**

**Answer :**

(i) ( – 4)^{5 }÷( – 4)^{8} = ( – 4)^{5 – 8} (a^{m} ÷ a^{n} = a^{m – n})

= ( – 4) ^{– 3
}

(ii)

(iii)

(iv) (3 ^{– 7} ÷ 3 ^{– 10}) × 3 ^{– 5} = (3 ^{– 7 – ( – 10)}) × 3^{ – 5} (a^{m} ÷ a^{n} = a^{m – n})

= 3^{3} × 3^{ – 5}

= 3^{3 + ( – 5)} (a^{m} × a^{n} = a^{m + n})

= 3 ^{– 2
}

(v) 2 ^{– 3} × ( – 7) ^{– 3} =

**Q3 : Find the value of.**

**(i) (3 ^{0} + 4 ^{– 1}) × 2^{2} **

**(ii) (2**

^{– 1}× 4^{– 1}) ÷2^{– 2}**(iii) (iv) (3**

^{– 1}+ 4^{– 1}+ 5^{– 1})^{0}**(v)**

**Answer :**

(i)

(ii) (2

^{ – 1}× 4

^{ – 1})

**÷**2

^{ – 2}= [2

^{ – 1}× {(2)

^{2}}

^{ – 1}]

**÷**2

^{ – 2}

= (2

^{ – 1}× 2

^{ – 2})

**÷**2

^{ – 2}= 2

^{ – 1+ ( – 2)}

**÷**2

^{ – 2}(

*a*

^{m}×

*a*

^{n}=

*a*

^{m}

^{+}

^{n})

= 2

^{ – 3}

**÷**2

^{ – 2}

= 2

^{ – 3}

^{– ( – 2)}(

*a*

^{m}

**÷**

*a*

^{n}=

*a*

^{m}

^{ – }

^{n})

= 2

^{ – 3 + 2}= 2

^{– 1}

(iii)

(iv) (3

^{– 1}+ 4

^{– 1}+ 5

^{– 1})

^{0}

= 1 (a

^{0}= 1)

(v)

**Q4 : Evaluate (i) (ii)**

**Answer :**

(i)

(ii)

**Q5 : Find the value of m for which 5^{m} ÷5^{-3} = 5^{5}.**

Answer :

5^{m} **÷** 5^{-3} = 5^{5}

5^{m} ^{– (- 3)} = 5^{5} (*a*^{m} **÷** *a*^{n} = *a*^{m}^{–} ^{n})

5^{m}^{+ 3} = 5^{5}

Since the powers have same bases on both sides, their respective exponents must be equal.

*m* + 3 = 5

*m* = 5 – 3

*m* = 2

**Q6 : Evaluate (i) (ii)**

**Answer :**

(i)

(ii)

**Q7 : Simplify. (i) (ii)**

**Answer :**

(i)

(ii)

**Exercise 12.2 :** Solutions of Questions on Page Number :** 200**

**Q1 :Express the following numbers in standard form.**

**(i) 0.0000000000085 (ii) 0.00000000000942**

**(iii) 6020000000000000 (iv) 0.00000000837**

**(v) 31860000000**

**Answer :**

(i) 0.0000000000085 = 8.5 x 10^{-12}

(ii) 0.00000000000942 = 9.42 x 10^{-12}

(iii) 6020000000000000 = 6.02 x 10^{15}

(iv) 0.00000000837 = 8.37 x 10^{-9}

(v) 31860000000 = 3.186 x 10^{10
}

**Q2 :Express the following numbers in usual form.**

**(i) 3.02 x 10 ^{-6} (ii) 4.5 x 10^{4}**

**(iii) 3 x 10**

^{-8}(iv) 1.0001 x 10^{9}**(v) 5.8 x 10**

^{12}(vi) 3.61492 x 10^{6}**Answer :**

(i) 3.02 x 10^{-6} = 0.00000302

(ii) 4.5 x 10^{4} = 45000

(iii) 3 x 10^{-8} = 0.00000003

(iv) 1.0001 x 10^{9} = 1000100000

(v) 5.8 x 10^{12} = 5800000000000

(vi) 3.61492 x 10^{6} = 3614920

**Q3 :Express the number appearing in the following statements in standard form.**

**(i) 1 micron is equal to m.**

**(ii) Charge of an electron is 0.000, 000, 000, 000, 000, 000, 16 coulomb.**

**(iii) Size of a bacteria is 0.0000005 m**

**(iv) Size of a plant cell is 0.00001275 m**

**(v) Thickness of a thick paper is 0.07 mm**

**Answer :**

(i) = 1 × 10^{ – 6}

(ii) 0.000, 000, 000, 000, 000, 000, 16 = 1.6 × 10^{ – 19}

(iii) 0.0000005 = 5 × 10^{ – 7}

(iv) 0.00001275 = 1.275 × 10^{ – 5}

(v) 0.07 = 7 × 10^{ – 2
}

**Q4 :In a stack there are 5 books each of thickness 20 mm and 5 paper sheets each of thickness 0.016 mm. What is the total thickness of the stack?**

**Answer :**

Thickness of each book = 20 mm

Hence, thickness of 5 books = (5 x 20) mm = 100 mm

Thickness of each paper sheet = 0.016 mm

Hence, thickness of 5 paper sheets = (5 x 0.016) mm = 0.080 mm

Total thickness of the stack = Thickness of 5 books + Thickness of 5 paper sheets

= (100 + 0.080) mm

= 100.08 mm

= 1.0008 x 10^{2} mm