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**Q1. Find the values of each of the following :**

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

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

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

**Sol:**

(i) 13^{2} = 13×13

= 169

(ii) 7^{3} = 7×7×7

= 4

(iii) 3^{4} = 3×3×3×3

= 81

**Q2. Find the value of each of the following :**

**(i) (−7) ^{2}**

**(ii) (−3) ^{4}**

**(iii) (−5) ^{5}**

**Sol:**

We know that if ‘a’ is a natural number, then

(−a) ^{even number} = positive number

(−a) ^{odd number} = negative number

We have,

(i) (−7)^{2} = (-7) ×(-7)

= 49

(ii) (−3)^{4} = (-3) ×(-3) ×(-3) ×(-3)

= 81

(iii) (−5)^{5} = (-5) ×(-5) ×(-5) ×(-5) ×(-5)

= -3125

**Q3. Simply :**

**(i) 3×10 ^{2}**

**(ii) 2 ^{2}×5^{3}**

**(iii) 3 ^{3}×5^{2}**

**Sol:**

(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

**Q4. Simply :**

**(i) 3 ^{2}×10^{4}**

**(ii) 2 ^{4}×3^{2}**

**(iii) 5 ^{2}×3^{4}**

**Sol:**

(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

**Q5. Simply :**

**(i) (−2)×(−3) ^{3}**

**(ii) (−3) ^{2}×(−5)^{3}**

**(iii) (−2) ^{5}×(−10)^{2}**

**Sol:**

(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

**Q6. Simply :**

**Sol:**

**Q7. 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}**

**Sol:**

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

= 2^{5} = 2×2×2×2×2

= 32

= 5^{2} = 5×5

= 25

Therefore, 2^{5} 5^{2}

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

= 3^{4} = 3×3×3×3

= 81

= 4^{3} = 4×4×4

= 64

Therefore, 3^{4} 4^{3}

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

= 3^{5} = 3×3×3×3×3

= 243

= 5^{3} = 5×5×5

= 125

Therefore, 3^{5} 5^{3}

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

= 3^{5} = 3×3×3×3×3

= 243

= 5^{3} = 5×5×5

= 125

Therefore, 3^{5} 5^{3}

**Q8. Express each of the following in exponential form**

**(i) (-5) ****×(-5) ×(-5)**

**Sol:**

**Q9. 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**

**Sol:**

(i) x ×x ×x ×x ×a ×a ×b ×b ×b = x^{4}a^{2}b^{3}

(ii) (-2) ×(-2) ×(-2) ×(-2) ×a×a×a = (−2)^{4}a^{3}

**Q10. Express each of the following numbers in exponential form**

**(i) 512**

**(ii) 625**

**(iii) 729**

**Sol:**

(i) 512 = 2^{9}

(iii) 625 = 5^{4}

(iii) 729 = 3^{6}

**Q11. Express each of the following numbers as a product of powers of their prime factors**

**(i) 36**

**(ii) 675**

**(iii) 392**

**Sol:**

(i) 36 = 2×2×3×3

= 2^{2}×3^{2}

(ii) 675 = 3×3×3×5×5

= 3^{3}×5^{2}

(iii) 392 = 2×2×2×7×7

= 2^{3}×7^{2}

**Q12. Express each of the following numbers as a product of powers of their prime factors**

**(i) 450**

**(ii) 2800**

**(iii) 24000**

**Sol:**

(i) 450 = 2×3×3×5×5

= 2×3^{2}×5^{2}

(ii) 2800 = 2×2×2×2×5×5×7

= 2^{4}×5^{2}×7

(iii) 24000 = 2×2×2×2×2×2×3×5×5×5

= 2^{5}×3×5^{3}

**Q13. Express each of the following as a rational number of the form p/q**

Sol:

**Q14. Express each of the following rational numbers in power notation**

Sol:

Because 7^{2} = 49 and 8^{2} = 64

Because 4^{3} = 64 and 5^{3} = 125

Because 1^{3} = 1 and 6^{3} = 216

**Q15. Find the value of the following**

**Sol:**

= 9/8

**Q16. If a= 2 and b = 3, the find the values of each of the followimg**

**(i) (a+b) ^{a}**

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

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

**Sol:**

(i) (a+b)^{a} = (2+3)^{2}

= (5)^{2}

= 25

(ii) (ab)^{b} = (2×3)^{3}

= (6)^{3}

= 216