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

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(ii) ((5² )³ × 5⁴) ÷ 5⁷
= [5⁶ × 5⁴] ÷ 5⁷ [(a^m)ⁿ = a^mn]
= 5^{6 + 4} ÷ 5⁷ [am × an = am + n]
= 5¹⁰ ÷ 5⁷
= 5^{10 - 7} [a^m ÷ aⁿ = a^{m - n}]
= 5³

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(iii) 25⁴ ÷ 5³
= (5²)⁴ ÷ 5³
= 5⁸ ÷ 5³ [(am)ⁿ = a^mn]
= 5^{8 - 3} [a^m ÷ aⁿ = a^{m - 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]
= \(\frac{7^2\times 11^8}{7\times11^3}\)
= 7^{2 - 1} × 11^{8 - 3} [a^m ÷ aⁿ = a^{m - n}]
= 7 × 11⁵

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

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(vi) 2⁰ + 3⁰ + 4⁰
= 1 + 1 + 1 [a^o = 1]
= 3
(vii) 2⁰ × 3⁰ × 4⁰
= 1 × 1 × 1 [a^o = 1]
= 1

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(viii) (3⁰ + 2⁰) × 5⁰
= (1 + 1) × 1 [a^o = 1]
= 2 × 1
= 2

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(ix) \(\frac{2^8\times a^5}{4^3\times a^3}\)
= \(\frac{2^8\times a^5}{2^{2^3\times a^3}}\)
= \(\frac{2^8\times a^5}{2^6\times a^3}\) [(a^m)ⁿ = a^mn]
= 2^{8 - 6} × a^{5 - 3} [a^m ÷ aⁿ = a^{m - n}]
= 2² × a²

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(x)\(\frac{a^5}{a^3}\times a^8\)
= a^{5 - 3} × a⁸ [a^m ÷ aⁿ = a^{m - n}]
= a² × a⁸
= a^{2 + 8} [a^m × aⁿ = a^{m + n}]
= a¹⁰

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(xi) \(\frac{4^5\times a^8b^3}{4^5\times a^5b^2}\)
= 4^{5 - 5} × a^{8 - 5} × b^{3 - 2} [a^m ÷ aⁿ = a^{m - n}]
= 4⁰ × a³ × b
= a³ × b [a^o = 1]

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(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