# Chapter – 12: Exponents and Power

## Exercise 12.1

**1. Evaluate:**

**(i) 3**^{-2 }**(ii) (-4)**^{-2 }**(iii) (1/2)**^{-5}** **

**Solution:**

(i) 3^{-2 }= (1/3)^{2}

= 1/9

(ii) **(-4)**^{-2 }**= (1/-4)**^{2}

= 1/16

(iii) **(1/2)**^{-5}** = (2/1)**^{5}

= 2^{5}

= 32

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

**(i) (-4)**^{4 }**÷(-4)**^{8}** **

**(ii) (1/2**^{3}**)**^{2}** **

**(iii) -(3)**^{4}**×(5/3)**^{4}** **

**(iv) (3**^{-7}**÷3**^{-10}**)×3**^{-5}** **

**(v) 2**^{-3}**×(-7)**^{-3}

**Solution:**

(i)

= (-4)^{5}/(-4)^{8}

= (-4)^{5-8}

= 1/(-4)^{3}

(ii) **(1/2**^{3}**)**^{2}** **

= 1^{2}/(2^{3})^{2}

= 1/2^{3×2} = 1/2^{6}

(iii) **-(3)**^{4}**×(5/3)**^{4}** **

= (-1)^{4}×3^{4}×(5^{4}/3^{4} )

= 3^{(4-4)}×5^{4}

= 3^{0}×5^{4} = 5^{4}

(iv)** **

= (3^{-7}/3^{-10})× 3^{-5}

= 3^{-7 – (-10) }× 3^{-5}

= 3^{(-7+10)}×3^{-5}

= 3^{3}×3^{-5}

= 3^{(3+-5)}

= 3^{-2}

=1/3^{2}

(v) 2^{-3}×(-7)^{ – 3}

= (2×-7)^{-3}

(Because a^{m}×b^{m} = (ab)^{m})

= 1/(2×-7)^{3}

= 1/(-14)^{3}

**3. Find the value of :**

**(i) (3**^{0}**+4**^{-1}**)×2**^{2}

**(ii) (2**^{-1}**×4**^{-1}**)÷2**^{ – 2}

**(iii) (1/2)**^{-2}**+(1/3)**^{-2}**+(1/4)**^{-2}

**(iv) (3**^{-1}**+4**^{-1}**+5**^{-1}**)**^{0}

**(v) {(-2/3)**^{-2}**}**^{2}

**Solution:**

(i)(3^{0}+4^{– 1})×2^{2} = (1+(1/4))×2^{2}

= ((4+1)/4 )×2^{2}

= (5/4)×2^{2}

= (5/2^{2})×2^{2}

= 5×2^{(2-2)}

= 5×2^{0}

= 5×1 = 5

(ii)(2^{-1}×4^{-1})÷2^{-2}

= [(1/2)×(1/4)] ÷(1/4)

= (1/2×1/2^{2} )÷ 1/4

= 1/2^{3}÷1/4

= (1/8)×(4)

= 1/2

(iii) **(1/2)**^{-2}**+(1/3)**^{-2}**+(1/4)**^{-2}

= (2^{-1})^{-2}+(3^{-1})^{-2}+(4^{-1})^{-2}

= 2^{(-1×-2)}+3^{(-1×-2)}+4^{(-1×-2)}

= 2^{2}+3^{2}+4^{2}

= 4+9+16

=29

(iv) (3^{-1}+4^{-1}+5^{-1})^{0}

= 1

(v) **{(-2/3)**^{-2}**}**^{2}** = (-2/3)**^{-2×2}

= (-2/3)^{-4}

= (-3/2)^{4}

= 81/16

**4. Evaluate**

**(i) (8**^{-1}**×5**^{3}**)/2**^{-4}

**(ii) (5**^{-1}**×2**^{-2}**)×6**^{-1}** **

**Solution:**

(i) (8^{-1}×5^{3})/2^{-4}

=

= 2×125 = 250

**(ii)** **(5**^{-1}**×2**^{-2}**)×6**^{-1}** **

= (1/10)×1/6

= 1/60

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

**Solution:**

5* ^{m}* ÷ 5

^{-3}= 5

^{5}

5^{(m-(-3) )} = 5^{5}

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

Comparing exponents both sides, we get

m+3 = 5

m = 5-3

m = 2

**6. Evaluate**

**(i) **

**(ii) **

**Solution:**

**(i)**

= 3-4

= -1

**(ii)**

=

=

= 512/125

**7. Simplify.**

**(i) **

**(ii) **

Solution:

**(i)**

=

**(ii)**

## Exercise 12.2

**1. Express the following numbers in standard form.**

**(i) 0.0000000000085**

**(ii) 0.00000000000942**

**(iii) 6020000000000000**

**(iv) 0.00000000837**

**(v) 31860000000**

**Solution:**

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

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

(iii) 6020000000000000 = 6020000000000000×(10^{15}/10^{15}) = 6.02×10^{15}

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

(v) 31860000000 = 31860000000×(10^{10}/10^{10}) = 3.186×10^{10}

**2.Express the following numbers in usual form.**

**(i) 3.02×10**^{-6}

**(ii) 4.5×10**^{4}

**(iii)3×10**^{-8}

**(iv)1.0001×10**^{9}

**(v) 5.8×10**^{12}

**(vi)3.61492×10**^{6}

**Solution:**

(i) 3.02**×**10^{-6} = 3.02/10^{6} = 0 .00000302

(ii) 4.5**×**10^{4} = 4.5**×**10000 = 45000

(iii) 3**×**10^{-8} = 3/10^{8} = 0.00000003

(iv) 1.0001**×**10^{9} = 1000100000

(v) 5.8**×**10^{12} = 5.8**×**1000000000000 = 5800000000000

(vi) 3.61492**×**10^{6 }= 3.61492**×**1000000 = 3614920

**3. Express the number appearing in the following statements in standard form.**

**(i) 1 micron is equal to 1/1000000 m.****(ii) Charge of an electron is 0.000, 000, 000, 000, 000, 000, 16 coulomb.****(iii) Size of 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**

**Solution :**

(i) 1 micron = 1/1000000

= 1/10^{6}

= 1×10^{-6}

(ii) Charge of an electron is 0.00000000000000000016 coulombs.

= 0.00000000000000000016**×**10^{19}/10^{19}

= 1.6**×**10^{-19 }coulomb

(iii) Size of bacteria = 0.0000005

= 5/10000000 = 5/10^{7} = 5×10^{-7} m

(iv) Size of a plant cell is 0.00001275 m

= 0.00001275×10^{5}/10^{5}

= 1.275×10^{-5}m

(v) Thickness of a thick paper = 0.07 mm

0.07 mm = 7/100 mm = 7/10^{2} = 7×10^{-2} mm

**4. 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?**

Solution:

Thickness of one book = 20 mm

Thickness of 5 books = 20×5 = 100 mm

Thickness of one paper = 0.016 mm

Thickness of 5 papers = 0.016×5 = 0.08 mm

Total thickness of a stack = 100+0.08 = 100.08 mm

= 100.08×10^{2}/10^{2} mm

mm

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