Simplify ( 2 5 ÷ 2 8 ) 5 × 2 − 5 . Options: A . &nbsp 1 2 10 B . &nbsp 1 2 20 C . &nbsp 1 2 ( − 15 ) D . &nbsp 1 2 ( − 5 )

Answer: Option B : B ( 2 5 ÷ 2 8 ) 5 × 2 − 5 = ( 2 5 2 8 ) 5 × 2 − 5 = ( 2 5 − 8 ) 5 × 2 − 5 ( ∵ a m a n = a m − n ) = ( 2 − 3 ) 5 × 2 − 5 = 2 − 15 × 2 − 5 ( ∵ ( a m ) n = a m n ) = 2 − 15 − 5 ( ∵ a m × a n = a m + n ) = 2 − 20 = 1 2 20 ( ∵ a − m = 1 a m )

Simplify (25÷28)5 × 2−5.

Question

Simplify ${(2^{5} \div 2^{8})}^{5}$ × $2^{- 5} .$

Options:

A .

\frac{1}{2^{10}}

B .

\frac{1}{2^{20}}

C .

\frac{1}{2^{(- 15)}}

D .

\frac{1}{2^{(- 5)}}

Answer: Option B
:
B

${(2^{5} \div 2^{8})}^{5} \times 2^{- 5} = {(\frac{2^{5}}{2^{8}})}^{5} \times 2^{- 5}$
$= {(2^{5 - 8})}^{5} \times 2^{- 5} (∵ \frac{a^{m}}{a^{n}} = a^{m - n})$
$= {(2^{- 3})}^{5} \times 2^{- 5}$
$= 2^{- 15} \times 2^{- 5} (∵ (a^{m})^{n} = a^{m n})$
$= 2^{- 15 - 5} (∵ a^{m} \times a^{n} = a^{m + n})$
$= 2^{- 20}$
$= \frac{1}{2^{20}} (∵ a^{- m} = \frac{1}{a^{m}})$