Question

Simplify :-
$a) \frac{(2⁵)² \times {7}^{3} }{ {8}^{3} \times 7}$
$b) \frac{ {3}^{5} \times {10}^{5} \times 25 }{ {5}^{7} \times {6}^{5} }$
$c) \frac{25 \times {5}^{2} \times {t}^{8} }{ {10}^{3} \times {t}^{4} }$
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Answer 1

Answer:-

$a) \frac{(2⁵)² \times {7}^{3} }{ {8}^{3} \times 7} = \frac{ {2}^{5 \times 2 } \times {7}^{3} }{(2³)³ \times 7}$

I By using (a²)³ = a²³

$= > \frac{ {2}^{10} \times {7}^{3} }{ {2}^{3 \times 3} \times 7 } = \frac{ {2}^{10} \times {7}^{3} }{ {2}^{9} \times 7 }$

I By using a³/a² = a³-²

$= > {2}^{10 - 9} \times {7}^{3 - 1} = {2}^{1} \times {7}^{2}$

$= > 2 \times 7 \times 7 = 98$

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$b) \frac{ {3}^{5} \times {10}^{5} \times 25 }{ {5}^{7} \times {6}^{5} }$

${10}^{5} = 2 \times 5,25 = {5}^{2} ,6 = 2 \times 3$

$\frac{ {3}^{5} \times {10}^{5} \times 25 }{ {5}^{7} \times {6}^{5} } = \frac{ {3}^{5} \times {(2 \times 5)}^{5} \times {5}^{2} }{ {5}^{7} \times {(2 \times 3)}^{5} }$

$= > \frac{ {3}^{5} \times {2}^{5} \times {5}^{5} \times {5}^{2} }{ {5}^{7} \times {2}^{5} \times {3}^{5} }$

I By using (ab)² = a²b²

$= > \frac{ {3}^{5} \times {2}^{5} \times {5}^{5 + 2} }{ {2}^{5} \times {3}^{5} \times {5}^{7} }$

$= > \frac{ {2}^{5} \times {3}^{5} \times {5}^{7} }{ {2}^{5} \times {3}^{5} \times {5}^{7} }$

$= > {2}^{5 - 5} \times {3}^{5 - 5} \times {5}^{7 - 7} = {2}^{0} \times {3}^{0} \times {5}^{0}$

I a⁰ = 1

$= > 1 \times 1 \times 1 = 1$

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$c) \frac{25 \times {5}^{2} \times {t}^{8} }{ {10}^{3} \times {t}^{4} } = \frac{ {5}^{2} \times {5}^{2} \times {t}^{8} }{ {(2 \times 5)}^{3} \times {t}^{4} }$

$= > \frac{ {5}^{2 + 2} \times {t}^{8} }{ {2}^{3} \times {5}^{3} \times {t}^{4} } = \frac{ {5}^{4} \times {t}^{8} }{ {2}^{3} \times {5}^{3} \times {t}^{4} }$

$= > \frac{ {5}^{4 - 3} \times {t}^{8 - 4} }{ {2}^{3} } = \frac{ {5}^{1} \times {t}^{4} }{ {2}^{3} } = \frac{ {5t}^{4} }{8}$