Question

1)      Investigate the relationship between a and b if - 2^a/2^b=32
2)      Given that 2^a/2^b=32            and           5^a/25=125

find the values of a and b.

               

Answer 1

Rules,
$\frac{a^{p}}{a^{q}} = a^{(p - q)} \\ \\ a^{p}a^{q} = a^{(p + q)}$

$\frac{ 2^{a} }{ 2^{b} } = 32 \\ \\ 2^{(a - b)} = 2^{5} \\ \\ a - b = 5$

$\frac{ 5^{a} }{25} = 125 \\ \\ 5^{a} = 25 * 125 \\ \\ 5^{a} = 5^{2} * 5^{3} \\ \\ 5^{a} = 5^{2 + 3} \\ \\ a = 5$

As a - b = 5, and a = 5 so, b = 0


Answer
a = 5, b = 0