**Exercise 6.1**

**Question 1: Find the values of each of the following:**

**(i) 13 ^{2}**

**(ii) 7 ^{3}**

**(iii) 3 ^{4} **

**Solution 1:**

(i) 13^{2}

= 13 × 13 =169

(ii) 7^{3}

= 7 × 7 × 7 = 343

(iii) 3^{4}

= 3 × 3 × 3 × 3 = 81

**Question 2: Find the value of each of the following:**

**(i) (-7) ^{2}**

**(ii) (-3) ^{4}**

**(iii) (-5) ^{5}**

**Solution 2:**

(i) (-7)^{2}

= (-7) × (-7)

= 49

(ii) (-3)^{4}

= (-3) × (-3) × (-3) × (-3)

= 81

(iii) (-5)^{5}

= (-5) × (-5) × (-5) × (-5) × (-5)

= -3125

**Question 3**: **Simplify:**

**(i) 3 × 10 ^{2}**

**(ii) 2 ^{2} × 5^{3}**

**(iii) 3 ^{3} × 5^{2}**

**Solution 3:**

(i) 3 × 10^{2}

= 3 × 10 × 10

= 3 × 100

= 300

(ii) 2^{2} × 5^{3}

= 2 × 2 × 5 × 5 × 5

= 4 × 125

= 500

(iii) 3^{3} × 5^{2}

= 3 × 3 × 3 × 5 × 5

= 27 × 25

= 675

**Question 4: Simply:**

**(i) 3 ^{2} × 10^{4}**

**(ii) 2 ^{4} × 3^{2}**

**(iii) 5 ^{2} × 3^{4}**

**Solution 4:**

(i) 3^{2 }× 10^{4}

= 3 × 3 × 10 × 10 × 10 × 10

= 9 × 10000

= 90000

(ii) 2^{4} × 3^{2}

= 2 × 2 × 2 × 2 × 3 × 3

= 16 × 9

= 144

(iii) 5^{2} × 3^{4}

= 5 × 5 × 3 × 3 × 3 × 3

= 25 × 81

= 2025

**Question 5: Simplify:**

**(i) (-2) × (-3) ^{3}**

**(ii) (-3) ^{2} × (-5)^{3}**

**(iii) (-2) ^{5} × (-10)^{2}**

**Solution 5:**

(i) (-2) × (-3)^{3}

= (-2) × (-3) × (-3) × (-3)

= (-2) × (-27)

= 54

(ii) (-3)^{2} × (-5)^{3}

= (-3) × (-3) × (-5) × (-5) × (-5)

= 9 × (-125)

= -1125

(iii) (-2)^{5} × (-10)^{2 }

= (-2) × (-2) × (-2) × (-2) × (-2) × (-10) × (-10)

= (-32) × 100

= -3200

**Question 6: Simplify:**

**(i) (3/4) ^{2}**

**(ii) ((-2)/3) ^{4 }**

**(iii) ((-4)/5) ^{5}**

**Solution 6:**

(i) (3/4)^{2 }

= (3/4) × (3/4)

= (9/16)

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

= ((-2)/3) × ((-2)/3) × ((-2)/3) × ((-2)/3)

= (16/81)

(iii) ((-4)/5)^{5}

= ((-4)/5) × ((-4)/5) × ((-4)/5) × ((-4)/5) × ((-4)/5)

= ((-1024)/3125)

**Question 7: Identify the greater number in each of the following:**

**(i) 2 ^{5} or 5^{2}**

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

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

**Solution 7:**

(i) 2^{5} or 5^{2}

2^{5} = 2 × 2 × 2 × 2 × 2 = 32

5^{2} = 5 × 5 = 25

So, 2^{5} > 5^{2}

(ii) 3^{4} or 4^{3}

3^{4} = 3 × 3 × 3 × 3 = 81

4^{3 }= 4 × 4 × 4 = 64

So, 3^{4} > 4^{3}

(iii) 3^{5 }or 5^{3}

3^{5} = 3 × 3 × 3 × 3 × 3 = 243

5^{3} = 5 × 5 × 5 = 125

So, 3^{5} > 5^{3}

**Question 8: Express each of the following in exponential form:**

**(i) (-5) × (-5) × (-5)**

**(ii) ((-5)/7) × ((-5)/7) × ((-5)/7) × ((-5)/7)**

**(iii) (4/3) × (4/3) × (4/3) × (4/3) × (4/3)**

**Solution 8:**

^{3}

^{4}

^{5}

**Question 9: Express each of the following in exponential form:**

**(i) x × x × x × x × a × a × b × b × b**

**(ii) (-2) × (-2) × (-2) × (-2) × a × a × a**

**(iii) (-2/3) × (-2/3) × x × x × x**

**Solution 9:**

^{4}a

^{2}b

^{3}

^{4}a

^{3}

^{2}x

^{3}

**Question 10: Express each of the following numbers in exponential form:**

**(i) 512**

**(ii) 625**

**(iii) 729**

**Solution 10:**

(i) 512

By the prime factorization of 512

= 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

= 2^{9}

(ii) 625

By the prime factorization of 625

= 5 × 5 × 5 × 5

= 5^{4}

(iii) 729

By the prime factorization of 729

= 3 × 3 × 3 × 3 × 3 × 3

= 3^{6}

**Question 11: Express each of the following numbers as a product of powers of their prime factors:**** (i) 36**** (ii) 675**** (iii) 392 **

**Solution 11:**

(i) 36

By the prime factorization of 36

= 2 × 2 × 3 × 3

= 2^{2} × 3^{2}

(ii) 675

By the prime factorization of 675

= 3 × 3 × 3 × 5 × 5

= 3^{3} × 5^{2}

(iii) 392

By the prime factorization of 392

= 2 × 2 × 2 × 7 × 7

= 2^{3} × 7^{2}

**Question 12: Express each of the following numbers as a product of powers of their prime factors:**** (i) 450**** (ii) 2800**** (iii) 24000**** **

**Solution 12:**

(i) 450

By the prime factorization of 450

= 2 × 3 × 3 × 5 × 5

= 2 × 3^{2} × 5^{2}

(ii) 2800

By the prime factorization of 2800

= 2 × 2 × 2 × 2 × 5 × 5 × 7

= 2^{4} × 5^{2} × 7

(iii) 24000

By the prime factorization of 24000

= 2 × 2 × 2 × 2 × 2 × 2 × 3 × 5 × 5 × 5

= 2^{6} × 3 × 5^{3}

** **

**Question 13: Express each of the following as a rational number of the form (p/q):**

**(i) (3/7)**

^{2}**(ii) (7/9)**

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

^{4}**Solution 13:**

^{2}

^{3}

^{4}

**Question 14: Express each of the following rational numbers in power notation:**

**(i) (49/64)**

**(ii) ((-64)/125)**

**(iii) ((-1)/216)**

**Solution 14:**

^{2}(7 ×7) = 49 and 8

^{2}(8 ×8) = 64

^{2}

^{3}(4 ×4) = 64 and 5

^{3}(5 × 5) = 125

^{3}

^{3}(1 ×1)= = 1 and 63 (6 ×6)= = 216

^{3}

** **

**Question 15: Find the value of the following:**

**(i) ((-1)/2)**

^{2}× 2^{3}× (3/4)^{2}**(ii) ((-3)/5)**

^{4}× (4/9)^{4}× ((-15)/18)^{2}**Solution 15:**

^{2}× 2

^{3}× (3/4)

^{2}

^{4}× (4/9)

^{4}× ((-15)/18)

^{2}

**Question 16: If a = 2 and b= 3, the find the values of each of the following:**

**(i) (a + b)**

^{a}**(ii) (a b)**

^{b}**(iii) (b/a)**

^{b}**(iv) 〖{(a/b)+(b/a)}〗**

^{a}

**Solution 16:**

^{a}

^{a }

^{2}

^{2}

^{b }

^{b }

^{3}

^{3}

^{b}

^{b}

^{3}

^{a}

^{a}

^{2}

**Exercise 6.2**

**Question 1: Using laws of exponents, simplify and write the answer in exponential form**

**(i) 2 ^{3} × 2^{4} × 2^{5}**

**(ii) 5 ^{12} ÷ 5^{3}**

**(iii) (7 ^{2})^{3}**

**(iv) (3 ^{2})^{5} ÷ 3^{4}**

**(v) 3 ^{7} × 2^{7}**

**(vi) (5 ^{21} ÷ 5^{13}) × 5^{7}**

**Solution 1:**

(i) 2^{3} × 2^{4} × 2^{5}

According to the law of exponents: a^{m} × a^{n }× a^{p} = a^{(m+n+p)}

So, the equation is written as

= 2^{(3 + 4 + 5)}

= 2^{12}

(ii) 5^{12} ÷ 5^{3}

According to the law of exponents: a^{m }÷ a^{n }= a^{m-n}

So, the equation is written as

= 5^{12 – 3}

= 5^{9}

(iii) (7^{2})^{3}

According to the law of exponents: (a^{m})^{n} = a^{m x n }

So the equation is written as

= 7^{2 x 3}

= 7^{6 }

(iv) (3^{2})^{5} ÷ 3^{4}

According to the law of exponents: (a^{m})^{n} = a^{m x n}

So the equation is written as

= 3^{2 x 5} ÷ 3^{4}

= 3 ^{10 }÷ 3^{4}

According to the law of exponents: a^{m }÷ a^{n }= a^{m-n}

= 3^{(10 – 4)}

= 3^{6}

(v) 3^{7} × 2^{7}

According to the law of exponents: a^{m} × b^{m} = (a × b)^{m}

So the equation is written as

= (3 × 2)^{7}

= 6^{7}

(vi) (5^{21} ÷ 5^{13}) × 5^{7}

According to the law of exponents: a^{m }÷ a^{n }= a^{m-n}

= 5^{(21 -13)} × 5^{7}

So the equation is written as

= 5^{8} × 5^{7}

According to the law of exponents: a^{m} × a^{n} = a^{(m +n)}

So the equation is written as

= 5^{(8+7)}

= 5^{15}

**Question 2: Simplify and express each of the following in exponential form:**

**(i) {(2 ^{3})^{4} × 2^{8}} ÷ 2^{12}**

**(ii) (8 ^{2} × 8^{4}) ÷ 8^{3}**

**(iii) (5 ^{7}/5^{2})× 5^{3 }**

**(iv) (5 ^{4}× x^{10}y^{5})/ (5^{4} × x^{7}y^{4})**

**Solution 2:**

(i) {(2^{3})^{4} × 2^{8}} ÷ 2^{12}

= {2^{12} × 2^{8}} ÷ 2^{12}

= 2^{(12 + 8)} ÷ 2^{12}

= 2^{20} ÷ 2^{12}

= 2^{ (20 – 12) }

=^{ } 2^{8}

(ii) (8^{2} × 8^{4}) ÷ 8^{3}

= 8^{(2 + 4)} ÷ 8^{3}

= 8^{6} ÷ 8^{3}

= 8^{(6-3)}

= 8^{3}

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

= 2^{9}

(iii) (5^{7}/5^{2}) × 5^{3}

= 5^{(7-2)} × 5^{3}

= 5^{5} × 5^{3}

= 5^{(5 + 3)}

= 5^{8}

(iv) (5^{4}× x^{10}y^{5})/ (5^{4} × x^{7}y^{4})

= (5^{4-4}× x^{10-7}y^{5-4})

= 5^{0}x^{3}y^{1 }[since 5^{0} = 1]

= 1x^{3}y

**Question 3: Simplify and express each of the following in exponential form:**

**(i) {(3 ^{2})^{3} × 2^{6}} × 5^{6}**

**(ii)(x/y) ^{12 }× y^{24} × (2^{3})^{4}**

**(iii)(5/2) ^{6} × (5/2)^{2}**

**(iv) (2/3) ^{5}× (3/5)^{5}**

**Solution 3:**

(i) {(3^{2})^{3} × 2^{6}} × 5^{6}

= {3^{6} × 2^{6}} × 5^{6}

= 6^{6} × 5^{6}

= 30^{6}

(ii) (x/y)^{12 }× y^{24} × (2^{3})^{4}

= (x^{12}/y^{12}) × y^{24} × 2^{12}

= x^{12 }× y^{24-12} × 2^{12}

= x^{12} × y^{12 }× 2^{12}

= (2xy)^{12}

(iii)(5/2)^{6} × (5/2)^{2}

= (5/2)6+2

= (5/2)8

(iv) (2/3)^{5}× (3/5)^{5}

= (2/5)^{5}

**Question 4: Write 9 × 9 × 9 × 9 × 9 in exponential form with base 3.**** **

**Solution 4:**

= 9 × 9 × 9 × 9 × 9

= (9)^{5} = (3^{2})^{5}

= 3^{10}

**Question 5: Simplify and write each of the following in exponential form:**

**(i) (25) ^{3} ÷ 5^{3}**

**(ii) (81) ^{5} ÷ (3^{2})^{5}**

**(iii) 9 ^{8} × (x^{2})^{5}/ (27)^{4} × (x^{3})^{2}**

**(iv) 3 ^{2} × 7^{8} × 13^{6}/ 21^{2} × 91^{3}**

**Solution 5:**

(i) (25)^{3} ÷ 5^{3}

= (5^{2})^{3} ÷ 5^{3}

= 5^{6 }÷ 5^{3}

= 5^{6 – 3}

= 5^{3}

(ii) (81)^{5} ÷ (3^{2})^{5}

= (81)^{5} ÷ 3^{10 }->81 = 3^{4}

= (3^{4})^{5} ÷ 3^{10}

= 3^{20} ÷ 3^{10}

= 3^{20-10}

= 3^{10}

(iii) 9^{8} × (x^{2})^{5}/ (27)^{4} × (x^{3})^{2}

= (3^{2})^{8} × (x^{2})^{5}/ (3^{3})^{4}× (x^{3})^{2}

= 3^{16} × x^{10}/3^{12} × x^{6}

= 3^{16-12 }× x^{10-6}

= 3^{4} × x^{4}

= (3x)^{4}

(iv) (3^{2} × 7^{8} × 13^{6})/ (21^{2} × 91^{3})

= (3^{2} × 7^{2}7^{6 }× 13^{6})/(21^{2}× 13^{3 }× 7^{3})

= (21^{2} × 7^{6} × 13^{6})/(21^{2}× 13^{3 }× 7^{3})

= (7^{6 }× 13^{6})/(13^{3 }× 7^{3})

= 91^{6}/91^{3}

= 91^{6-3}

= 91^{3}

**Question 6: Simplify:**

**(i) (3 ^{5})^{11} × (3^{15})^{4} – (3^{5})^{18} × (3^{5})^{5}**

**(ii) (16 × 2 ^{n+1 }– 4 × 2^{n})/(16 × 2^{n+2 }– 2 × 2^{n+2})**

**(iii) (10 × 5 ^{n+1 }+ 25 × 5^{n})/(3 × 5^{n+2 }+ 10 × 5^{n+1})**

**(iv) (16) ^{7 }×(25)^{5}× (81)^{3}/(15)^{7 }×(24)^{5}× (80)^{3}**

**Solution 6:**

(i) (3^{5})^{11} × (3^{15})^{4} – (3^{5})^{18} × (3^{5})^{5}

= (3)^{55} × (3)^{60} – (3)^{90} × (3)^{25}

= 3 ^{55+60} – 3^{90+25}

= 3^{115} – 3^{115}

= 0

(ii) (16 × 2^{n+1 }– 4 × 2^{n})/(16 × 2^{n+2 }– 2 × 2^{n+2})

= (2^{4} × 2^{(n+1) }-2^{2} × 2^{n})/(2^{4} × 2^{(n+2) }-2^{n+1} × 2^{2})

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

= 2^{n} × 2^{3} – 2^{n}/ 2^{n} × 2^{4} – 2^{n }× 2

= 2^{n}(2^{3} – 1)/ 2^{n}(2^{4} – 1)

= 8 -1 /16 -2

= 7/14

= (1/2)

(iii) (10 × 5^{n+1 }+ 25 × 5^{n})/(3 × 5^{n+2 }+ 10 × 5^{n+1})

= (10 × 5^{n+1 }+ 5^{2} × 5^{n})/(3 × 5^{n+2 }+ (2 × 5) × 5^{n+1})

= (10 × 5^{n+1 }+ 5 × 5^{n+1})/(3 × 5^{n+2 }+ (2 × 5) × 5^{n+1})

= 5^{n+1} (10+5)/ 5^{n+1 }(10+15)

= 15/25

= (3/5)

(iv) (16)^{7 }×(25)^{5}× (81)^{3}/(15)^{7 }×(24)^{5}× (80)^{3}

= (16)^{7 }× (5^{2})^{5}× (3^{4})^{3}/(3 × 5 )^{7 }×(3 × 8)^{5}× (16 × 5)^{3}

= (16)^{7 }× (5^{2})^{5}× (3^{4})^{3}/3^{7} × 5^{7 }× 3^{5} × 8^{5}× 16^{3}× 5^{3}

= (16)^{7}/ 8^{5} × 16 ^{3}

= (16)^{4}/8^{5}

= (2 × 8)^{4}/8^{5}

= 2^{4}/8

= (16/8)

= 2

**Question 7: Find the values of n in each of the following:**

**(i) 5 ^{2n} × 5^{3} = 5^{11}**

**(ii) 9 × 3 ^{n} = 3^{7}**

**(iii) 8 × 2 ^{n+2} = 32 **

**(iv) 7 ^{2n+1} ÷ 49 = 7^{3}**

**(v) (3/2) ^{4} × (3/2) ^{5 }= (3/2)^{2n+1}**

**(vi) (2/3) ^{10}× {(3/2)^{2}}^{5} = (2/3)^{2n – 2}**

**Solution:**

(i) 5^{2n} × 5^{3} = 5^{11}

= 5^{2n+3 }= 5^{11}

On equating the coefficients, we find:

2n + 3 = 11

⇒2n = 11- 3

⇒2n = 8

⇒ n = (8/2)

⇒ n = 4

(ii) 9 × 3^{n} = 3^{7}

= (3)^{2} × 3^{n} = 3^{7}

= (3)^{2+n }= 3^{7}

On equating the coefficients, we find:

2 + n = 7

⇒ n = 7 – 2 = 5

(iii) 8 × 2^{n+2} = 32

= (2)^{3} × 2^{n+2} = (2)^{5} [since 2^{3} = 8 and 2^{5} = 32]

= (2)^{3+n+2 }= (2)^{5}

On equating the coefficients, we find:

3 + n + 2 = 5

⇒ n + 5 = 5

⇒ n = 5 -5

⇒ n = 0

(iv) 7^{2n+1} ÷ 49 = 7^{3}

= 7^{2n+1} ÷ 7^{2} = 7^{3} [since 49 = 7^{2}]

= 7^{2n+1-2} = 7^{3}

= 7^{2n-1}=7^{3}

On equating the coefficients, we find:

2n – 1 = 3

⇒ 2n = 3 + 1

⇒ 2n = 4

⇒ n =4/2 =2

(v) (3/2)^{4} × (3/2) ^{5 }= (3/2)^{2n+1}

= (3/2)^{4+5} = (3/2)^{2n+1}

= (3/2)^{9} = (3/2)^{2n+1}

On equating the coefficients, we find:

2n + 1 = 9

⇒ 2n = 9 – 1

⇒ 2n = 8

⇒ n = 8/2

=4

(vi) (2/3)^{10}× {(3/2)^{2}}^{5} = (2/3)^{2n – 2}

= (2/3)^{10} × (3/2)^{10} = (2/3)^{2n – 2}

= 2 ^{10} × 3^{10}/3^{10} × 2^{10} = (2/3)^{2n – 2}

= 1 = (2/3)^{2n – 2}

= (2/3)^{0} = (2/3)^{2n – 2}

On equating the coefficients, we find:

0 =2n -2

2n -2 =0

2n =2

n = 1

**Question 8: If (9 ^{n} × 3^{2} × 3^{n} – (27)^{n})/ (3^{3})^{5} × 2^{3} = (1/27), find the value of n.**

**Solution 8:**

= (9^{n} × 3^{2} × 3^{n} – (27)^{n})/ (3^{3})^{5} × 2^{3} = (1/27)

= (3^{2})^{n} × 3^{3} × 3^{n }– (3^{3})^{n}/ (3^{15} × 2^{3}) = (1/27)

= 3^{(2n+2+n)} – (3^{3})^{n}/ (3^{15} × 2^{3}) = (1/27)

= 3^{(3n+2)}– (3^{3})^{n}/ (3^{15} × 2^{3}) = (1/27)

= 3^{3n} × 3^{2} – 3^{3n}/ (3^{15} × 2^{3}) = (1/27)

= 3^{3n} × (3^{2} – 1)/ (3^{15} × 2^{3}) = (1/27)

= 3^{3n} × (9 – 1)/ (3^{15} × 2^{3}) = (1/27)

= 3^{3n} × (8)/ (3^{15} × 2^{3}) = (1/27)

= 3^{3n} × 2^{3}/ (3^{15} × 2^{3}) = (1/27)

= 3^{3n}/3^{15} = (1/27)

= 3^{3n-15} = (1/27)

= 3^{3n-15} = (1/3^{3})

= 3^{3n-15} = 3^{-3}

On equating the coefficients, we find:

3n -15 = -3

⇒ 3n = -3 + 15

⇒ 3n = 12

⇒ n = 12/3 = 4** **

**Exercise 6.3**

**Question 1: Express the following numbers in the standard form:**** (i) 3908.78**** (ii) 5,00,00,000**** (iii) 3,18,65,00,000**** (iv) 846 × 10 ^{7}**

**(v)723 × 10**

^{9}

**Solution 1:**

(i) 3908.78

= 3.90878 × 10^{3} [Here the decimal point is moved left up to 3 places]

(ii) 5,00,00,000

= 5,00,00,000.00 = 5 × 10^{7} [Here, the decimal point is moved left up to 7 places]

(iii) 3,18,65,00,000

= 3,18,65,00,000.00

= 3.1865 × 10^{9} [here, the decimal point is moved left up to 9 places]

(iv) 846 × 10^{7}

= 8.46 × 10^{2} × 10 [here the decimal point is moved left up to 2 places]

= 8.46 × 10^{9} [since a^{m} × a^{n} = a^{m+n}]

(v) 723 × 10^{9}

= 7.23 × 10^{2} × 10^{9} [here the decimal point is moved left up to 2 places]

= 7.23 × 10^{11} [ since a^{m} × a^{n} = a^{m+n}]

**Question 2: Write the following numbers in the usual form:**** (i) 4.83 × 10 ^{7}**

**(ii) 3.21 × 10**

^{5}**(iii) 3.5 × 10**

^{3}**Solution 2:**

(i) 4.83 × 10^{7}

= 483 × 10^{7-2} [hence the decimal point is moved right up to 2 places]

= 483 × 10^{5}

= 4, 83, 00,000

(ii) 3.21 × 10^{5 }

= 321 × 10^{5-2} [hence the decimal point is moved right up to 2 places]

= 321 × 10^{3}

= 3, 21,000

(iii) 3.5 × 10^{3}

= 35 × 10^{3-1} [hence the decimal point is moved right up to 1 places]

= 35 × 10^{2}

= 3,500

**Question :3. Express the numbers appearing in the following statements in the standard form:**

**(i) The distance between the Earth and the Moon is 384,000,000 meters.**** (ii) Diameter of the Earth is 1, 27, 56,000 meters.**** (iii) Diameter of the Sun is 1,400,000,000 meters.**** (iv) The universe is estimated to be about 12,000,000,000 years old.**** **

**Solution:**

(i) The distance between the Earth and the Moon is 384,000,000 meters.

Distance between the Earth and the Moon is 3.84 × 10^{8} meters.

[Here the decimal point is moved left up to 8 places.]

(ii) Diameter of the Earth is 1, 27, 56,000 meters.

Diameter of the Earth is 1.2756 × 10^{7} meters.

[Here the decimal point is moved left up to 7 places]

(iii) Diameter of the Sun is 1,400,000,000 meters.

Diameter of the Sun is 1.4 × 10^{9} meters.

[Here the decimal point is moved left up to 9 places]

(iv) The universe is estimated to be about 12,000,000,000 years old.

The universe is estimated to be about 1.2× 10^{10} years old.

[Here the decimal point is moved left up to 10 places]

**Exercise 6.4**

**Question 1: Write the following numbers in the expanded exponential forms:**** (i) 20068**** (ii) 420719**** (iii) 7805192**** (iv) 5004132**** (v) 927303**** **

**Solution 1:**

(i) 20068

= 2 × 10^{4} + 0 × 10^{3} + 0 × 10^{2} + 6 × 10^{1} + 8 × 10^{0}

(ii) 420719

= 4 × 10^{5} + 2 × 10^{4} + 0 × 10^{3} + 7 × 10^{2} + 1 × 10^{1} + 9 × 10^{0}

(iii) 7805192

= 7 × 10^{6} + 8 × 10^{5} + 0 × 10^{4} + 5 × 10^{3} + 1 × 10^{2} + 9 × 10^{1} + 2 × 10^{0}

(iv) 5004132

= 5 × 10^{6} + 0 × 10^{5} + 0 × 10^{4} + 4 × 10^{3} + 1 × 10^{2} + 3 × 10^{1} + 2 × 10^{0}

(v) 927303

= 9 × 10^{5} + 2 × 10^{4} + 7 × 10^{3} + 3 × 10^{2} + 0 × 10^{1} + 3 × 10^{0}

**Question 2: Find the number from each of the following expanded forms:**** (i) 7 × 10 ^{4} + 6 × 10^{3} + 0 × 10^{2} + 4 × 10^{1} + 5 × 10^{0}**

**(ii) 5 × 10**

^{5}+ 4 × 10^{4}+ 2 × 10^{3}+ 3 × 10^{0}**(iii) 9 × 10**

^{5}+ 5 × 10^{2}+ 3 × 10^{1}**(iv) 3 × 10**

^{4 }+ 4 × 10^{2}+ 5 × 10^{0}

**Solution 2:**

(i) 7 × 10^{4} + 6 × 10^{3} + 0 × 10^{2} + 4 × 10^{1} + 5 × 10^{0}

= 7 × 10000 + 6 × 1000 + 0 × 100 + 4 × 10 + 5 × 1

= 70000 + 6000 + 0 + 40 + 5

= 76045

(ii) 5 × 10^{5} + 4 × 10^{4} + 2 × 10^{3} + 3 × 10^{0}

= 5 × 100000 + 4 × 10000 + 2 × 1000 + 3 × 1

= 500000 + 40000 + 2000 + 3

= 542003

(iii) 9 × 10^{5} + 5 × 10^{2} + 3 × 10^{1}

= 9 × 100000 + 5 × 100 + 3 × 10

= 900000 + 500 + 30

= 900530

(iv) 3 × 10^{4 }+ 4 × 10^{2} + 5 × 10^{0}

= 3 × 10000 + 4 × 100 + 5 × 1

= 30000 + 400 + 5

= 30405