Question

Simplify and express each of the following in exponential form:
(i) $\frac{2^{3} \times 3^{4} \times 4}{3 \times 32}$ (ii) $[{5^{2}}^{3} \times 5^{4}] \div 5^{7}$ (iii) $25^{4} \div 5^{3}$
(iv) $\frac{3 \times 7^{2} \times 11^{8}}{21 \times 11^{3}}$ (v) $\frac{3^{7}}{3^{4 \times} 3^{3}}$ (vi) 2⁰ + 3⁰ + 4⁰
(vii) 2⁰ × 3⁰ × 4⁰ (viii) (3⁰ + 2⁰) × 5⁰ (ix) $\frac{2^{8} \times a^{5}}{4^{3} \times a^{3}}$
(x) $(\frac{a^{5}}{a^{3}}) \times a^{8}$ (xi) $\frac{4^{5} \times a^{8} b^{3}}{4^{5} \times a^{5} \times b^{2}}$ (xii) ${(2^{3} \times 2)}^{2}$

Solution

(i)

$\frac{2^{3} \times 3^{4} \times 4}{3 \times 32} = \frac{2^{3} \times 3^{4} \times 2 \times 2}{3 \times 2 \times 2 \times 2 \times 2 \times 2} = \frac{2^{3} \times 3^{4} \times 2^{2}}{3 \times 2^{5}} = \frac{2^{3 + 2} \times 3^{4}}{3 \times 2^{5}} = \frac{2^{5} \times 3^{4}}{3 \times 2^{5}} = 2^{5 - 5} \times 3^{4 - 1} = 2^{0} 3^{0} = 1 \times 3^{3} = 3^{3}$

(ii) [(5²)³ × 5⁴] ÷ 5⁷

= [5^{2 × 3} × 5⁴] ÷ 5⁷ (a^m)ⁿ = a^mn

= [5⁶ × 5⁴] ÷ 5⁷

= [5^{6 + 4}] ÷ 5⁷ (a^m × aⁿ = a^m⁺ⁿ)

= 5¹⁰ ÷ 5⁷

= 5^{10 − 7} (a^m ÷ aⁿ = a^m⁻ⁿ)

= 5³

(iii) 25⁴ ÷ 5³ = (5 ×5)⁴ ÷ 5³

= (5²)⁴ ÷ 5³

= 5^{2 × 4} ÷ 5³ (a^m)ⁿ = a^mn

= 5⁸ ÷ 5³

= 5^{8 − 3} (a^m ÷ aⁿ = a^m⁻ⁿ)

= 5⁵

(iv)

$\frac{3 \times 7^{2} \times 11^{8}}{21 \times 11^{3}} = \frac{3 \times 7^{2} \times 11^{8}}{3 \times 7 \times 11^{3}} = 3^{1 - 1} \times 7^{2 - 1} \times 11^{8 - 3} = 3^{0} \times 7^{1} \times 11^{5}$

= 1 × 7 × 11⁵ = 7 × 11⁵

(v)

$\frac{3^{7}}{3^{4} \times 3^{3}} = \frac{3^{7}}{3^{4 + 3}} = \frac{3^{7}}{3^{7}} = 3^{7 - 7} = 3^{0} = 1$

(vi) 2⁰ + 3⁰ + 4⁰ = 1 + 1 + 1 = 3

(vii) 2⁰ × 3⁰ × 4⁰ = 1 × 1 × 1 = 1

(viii) (3⁰ + 2⁰) × 5⁰ = (1 + 1) × 1 = 2

(ix)

$\frac{2^{8} \times a^{5}}{4^{3} \times a^{3}} = \frac{2^{8} \times a^{5}}{{(2 \times 2)}^{3} \times a^{3}} = \frac{2^{8} \times a^{5}}{(2}$ ²)3×a3=28×a5(22×3)×a3= 28×a526×a3 = 28-6×a5-3=22×a2 = (2×a)2 = (2a)2

(x)

$(\frac{a^{5}}{a^{3}}) \times a^{8} = a^{5 - 3} \times a^{8} = a^{2} \times a^{8} = a^{2 + 8} = a^{10}$

(xi)

$\frac{4^{5} \times a^{8} b^{8}}{4^{5} \times a^{5} b^{2}} = 4^{5 - 5} \times a^{8 - 5} \times b^{3 - 2} = 4^{0} \times a^{3} \times b^{1} = 1 \times a^{3} \times b = a^{3} b$

(xii) (2³ × 2)² = ${(2^{3 + 1})}^{2}$ (a^m × aⁿ = a^m⁺ⁿ)

= (2⁴)² = 2^{4 × 2} (a^m)ⁿ = a^mn

= 2⁸

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Exponents And Powers

Some More Questions From Exponents and Powers Chapter

Express each of the following as product of powers of their prime factors:
(i) 648 (ii) 405
(iii) 540 (iv) 3,600

Simplify:
(i) 2 × 10³ (ii) 7² × 2²
(iii) 2³ × 5 (iv) 3 × 4⁴
(v) 0 × 10² (vi) 5² × 3³
(vii) 2⁴× 3² (viii) 3² × 10⁴

Simplify:
(i) (− 4)³
(ii) (− 3) × (− 2)³
(iii) (− 3)² × (− 5)²
(iv)(− 2)³ × (−10)³

Compare the following numbers:
(i) 2.7 × 10¹²; 1.5 × 10⁸
(ii) 4 × 10¹⁴; 3 × 10¹⁷

Say true or false and justify your answer:
(i) 10 × 10¹¹ = 100¹¹ (ii) 2³ > 5²
(iii) 2³ × 3² = 6⁵(iv) 3⁰ = (1000)⁰

Express each of the following as a product of prime factors only in exponential form:
(i) 108 × 192 (ii) 270
(iii) 729 × 64 (iv) 768

Write the following numbers in the expanded forms:
279404, 3006194, 2806196, 120719, 20068

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Exponents And Powers

Simplify and express each of the following in exponential form:(i) 23×34×43×32 (ii) 523×54÷57 (iii) 254÷53(iv) 3×72×11821×113 (v) 3734×33 (vi) 20 + 30 + 40(vii) 20 × 30 × 40 (viii) (30 + 20) × 50 (ix) 28×a543×a3(x) a5a3×a8 (xi) 45×a8b345×a5×b2 (xii) 23×22

Some More Questions From Exponents and Powers Chapter

Express each of the following as product of powers of their prime factors:(i) 648 (ii) 405(iii) 540 (iv) 3,600

Simplify:(i) 2 × 103 (ii) 72 × 22(iii) 23 × 5 (iv) 3 × 44(v) 0 × 102 ­­­­­ (vi) 52 × 33(vii) 24 × 32 (viii) 32 × 104

Simplify:(i) (− 4)3 (ii) (− 3) × (− 2)3(iii) (− 3)2 × (− 5)2 (iv)(− 2)3 × (−10)3

Compare the following numbers:(i) 2.7 × 1012; 1.5 × 108(ii) 4 × 1014; 3 × 1017

Say true or false and justify your answer:(i) 10 × 1011 = 10011 (ii) 23 > 52(iii) 23 × 32 = 65 (iv) 30 = (1000)0

Express each of the following as a product of prime factors only in exponential form:(i) 108 × 192 (ii) 270(iii) 729 × 64 (iv) 768

Write the following numbers in the expanded forms:279404, 3006194, 2806196, 120719, 20068