Q:

solve the exponential equation by rewriting the base. explain the steps towards your answer.[tex]32^{-2x} =16[/tex]

Accepted Solution

A:
Answer:The answer is x = -0.4Step-by-step explanation:* In the exponential functions we have some rules1- b^m  ×  b^n  =  b^(m + n) ⇒ in multiplication if they have same base we add  the power2- b^m  ÷  b^n =  b^(m – n) ⇒  in division if they have same base we subtract  the power3- (b^m)^n = b^(mn) ⇒ if we have power over power we multiply them4- a^m × b^m = (ab)^m ⇒ if we multiply different bases with same  power then we multiply them ad put over the answer the power5- b^(-m) = 1/(b^m)  (for all nonzero real numbers b) ⇒ If we have negative power we reciprocal the base to get positive power6- If  a^m  =  a^n  ,  then  m  =  n ⇒ equal bases get equal powers7- If  a^m  =  b^m  ,  then  a  =  b    or    m  =  0* Now lets solve our problem∵ 32^(-2x) = 16∵ 32 = 2^5 ⇒ 32 ÷ 2 =16 ÷ 2 = 8 ÷ 2 = 4 ÷ 2 = 2 ÷ 2 = 1 ∵ 16 = 2^4 ⇒ 16 ÷ 2 = 8 ÷ 2 = 4 ÷ 2 = 2 ÷ 2 = 1 ∵ (2^5)^(-2x) = 2^(5 × -2x) = 2^(-10x) ⇒ by using rule 3∴ 2^(-10x) = 2^4 ⇒ by using rule 6∴ -10x = 4 ⇒ divided by -10 for both sides∴ x = 4/-10 = -2/5 = -0.4* The answer is x = -0.4