Question

Simplify:
(i) \(\frac{(2^5)^2×7^3}{8^3×7}\) (ii) \(\frac{2^5×5^2×t^8}{10^3×t^4}\) (iii) \(\frac{3^5×10^5×25}{5^7×6^5}\)

Answer 1

(i)\(\frac{(2^5)^2×7^3}{8^3×7}\)
= \(\frac{(2^5)^2×7^3}{{(2×2×2)}^3×7}\) [(a^m)ⁿ=a^mn]
= \(\frac{2^{10}×7^3}{(2^3)^3×7}\)  [(a^m)ⁿ=a^mn]
= 2¹×7²=2×7×7=98

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(ii)\(\frac{2^5×5^2×t^8}{10^3×t^4}\)
= \(\frac{5\times5×5^2×t^8}{{(5\times 2)}^3×t^4}\) (a×b)^m=(a^m×b^m)
= \(\frac{5^{1+1+2}×t^8}{5^3×2^3×t^4}\) (a^m×aⁿ=a^m+n)
= \(\frac{5^{4-3}×t^{8-4}}{2^3}\) (a^m÷aⁿ=a^m-n)
= \(\frac{5^1×t^4}{2×2×2}\)
= \(\frac{5t^4}{8}\)

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(iii)\(\frac{3^5×10^5×25}{5^7×6^5}\)
= \(\frac{3^5×2^5×5^5×5^2}{5^7×2^5×3^5}\) (a×b)^m=(a^m×b^m)
= \(\frac{3^2×2^5×5^{5+2}}{5^7×2^5×3^5}\) (a^m×aⁿ=a^m+n)
= \(\frac{3^5×2^5×5^7}{5^7×2^5×3^5}\)
= 3^5-5×2^5-5×5^7-7 (a^m÷aⁿ=a^m-n)
= 3⁰×2⁰×5⁰
= 1×1×1
= 1