Question

( 2 ^5 ÷ 2^ 8 ) ^ 5 × 2 ^- 5​

Answer 1

Question:-

Solve the following:-

$\\ \bullet \: \: \sf \bigg( \dfrac{ {2}^{5} }{ {2}^{8} } \bigg) ^{5} \times {2}^{ - 5}$

Solution:-

We know that :-

$\\ \rightarrow \large\underline{ \boxed{ \sf \frac{ {a}^{m} }{ {a}^{n} } = {a}^{m - n} }} \bigstar$

Applying the same concept in the given expression:-

$\\ : \implies \sf \bigg({2}^{5 - 8} \bigg) ^{5} \times {2}^{ - 5}$

$\\ : \implies \sf \bigg( {2}^{ - 3} \bigg) ^{5} \times {2}^{ - 5}$

We also know that :-

$\\\large \rightarrow \underline{ \boxed{ \sf \bigg({a}^{m} \bigg) ^{n} = {a}^{m.n} }} \bigstar$

$\\ \large\rightarrow \underline{ \boxed{ \sf {a}^{m} \times {a}^{n} = {a}^{m + n} }} \bigstar$

Applying these two concepts :-

$\\ : \implies \sf {2}^{ - 15} \times {2}^{ - 5}$

$\\ : \implies \sf {2}^{ - 15 + ( - 5)}$

$\\ : \implies \sf {2}^{ - 15 - 5}$

$\\ : \implies \sf {2}^{ - 20}$

$\\ : \implies \boxed{ \boxed{\sf \frac{1}{ {2}^{20} } }} \bigstar \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bigg \{ \sf{ {a}^{ - m} = \frac{1}{ {a}^{m} } } \bigg \}$

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hope it helps