Question

Simplify and express each of the following in exponential form:
(i) $\frac{2^3\times 3^4\times4}{3\times 3^2}$
(ii) ((52)3×54)÷57
(iii) 254÷53
(iv) $\frac{3\times7^2\times 11^8}{21\times11^3}$
(v) $\frac{3^7\times3^4}{3^3}$
(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) $\frac{4^5\times a^8b^3}{4^5\times a^5b^2}$
(xii) (23×2)2

Answer 1

(i) $\frac{2^3\times 3^4\times4}{3\times 3^2}$
= $\frac{2^3\times 3^4\times2^2}{3\times 3^2}$
= $\frac{2^5\times 3^4-1}{2^5}$ [am ÷ an = am - n]
= 33

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(ii) ((52 )3 × 54) ÷ 57
= [56 × 54] ÷ 57 [(am)n = amn]
= 56 + 4 ÷ 57 [am × an = am + n]
= 510 ÷ 57
= 510 - 7 [am ÷ an = am - n]
= 53

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(iii) 254 ÷ 53
= (52)4 ÷ 53
= 58 ÷ 53 [(am)n = amn]
= 58 - 3 [am ÷ an = am - n]
= 55

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(iv) $\frac{3\times7^2\times 11^8}{21\times11^3}$
= $\frac{3\times7^2\times 11^8}{3\times7\times11^3}$ [Since, 21 = 3 × 7]
= $\frac{7^2\times 11^8}{7\times11^3}$
= 72 - 1 × 118 - 3 [am ÷ an = am - n]
= 7 × 115

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(v) $\frac{3^7\times3^4}{3^3}$
= 37 / 34 + 3 [am × an = am + n]
= $\frac{3^7}{3^7}$
= 37 - 7 [am ÷ an = am-n]
= 30
= 1 [ao = 1]

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(vi) 20 + 30 + 40
= 1 + 1 + 1 [ao = 1]
= 3
(vii) 20 × 30 × 40
= 1 × 1 × 1 [ao = 1]
= 1

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(viii) (30 + 20) × 50
= (1 + 1) × 1 [ao = 1]
= 2 × 1
= 2

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(ix) $\frac{2^8\times a^5}{4^3\times a^3}$
= $\frac{2^8\times a^5}{2^{2^3\times a^3}}$
= $\frac{2^8\times a^5}{2^6\times a^3}$ [(am)n = amn]
= 28 - 6 × a5 - 3 [am ÷ an = am - n]
= 22 × a2

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(x)$\frac{a^5}{a^3}\times a^8$
= a5 - 3 × a8 [am ÷ an = am - n]
= a2 × a8
= a2 + 8 [am × an = am + n]
= a10

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(xi) $\frac{4^5\times a^8b^3}{4^5\times a^5b^2}$
= 45 - 5 × a8 - 5 × b3 - 2 [am ÷ an = am - n]
= 40 × a3 × b
= a3 × b [ao = 1]

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(xii) (23 × 2)2
= (23 + 1)2 [am × an = am + n]
= (24)2
= 24 × 2 [(am)n = amn]
= 28