Question:

Simplify.
  1.  (25×t4)(53×10×t8)(t0)\frac{(25×t^{−4})}{(5^{−3}×10×t^{−8})} (t ≠ 0)
  2. (35×105×125)(57×65)\frac{ (3^{−5}×10^{−5}×125)}{(5^{−7}×6^{−5})}

Updated On: Dec 2, 2023
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Solution and Explanation

(i) (25×t4)(53 ×10×t8)\frac{(25 × t^{−4})}{(5^{−3} × 10 × t^{−8})}

=(52 ×t4)(53 ×5×2×t8 )= \frac{(5^2 × t^{−4})}{(5^{−3} × 5 × 2 × t^{−8} )}

=(52 ×t4)(53+1 ×2×t8)= \frac{(5^2 × t^{−4})}{(5^{−3 + 1} × 2 × t^{−8}) }        [Since, a× a= am + n]

=(52 ×t4)(52 ×2×t8)= \frac{(5^2 × t^{−4})}{(5^{−2} × 2 × t^{−8})}

=(52(2) ×t4(8))2 = \frac{(5^{2−(−2)} × t^{−4−(−8)})}{2}                     [Since, aman =amn\frac{a^m}{a^n} = a^{m − n}]
=(54 ×t4+8)2= \frac{(5^4 × t^{−4 + 8})}{2}

=625 t42=\frac{ 625\ t^4}{2}


(ii) (35×105×125)(57×65)\frac{(3^{−5}×10^{−5}×125)}{(5^{−7}× 6^{−5})}

=(35 ×(2×5)5 ×53)(57×(2×3)5)= \frac{(3^{−5} × (2 × 5)^{−5} × 5^3)}{(5^{−7}× (2 × 3)^{-5})}

=35(5) ×25(5) ×55(7)+3 = 3^{−5−(−5)} × 2^{−5−(−5)} × 5^{−5−(−7)+3  }  [Since, am ×an =am+na^m × a^n = a^{m + n} and aman =amn\frac{a^m}{a^n} = a^{m − n}]
=30 ×20 ×55= 3^0 × 2^0 × 5^5
=1×1×55= 1 × 1 × 5^5     [a0=1][∵a^0=1]
=55= 5^5

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