Answer:
The answer is x = -0.4
Step-by-step explanation:
* In the exponential functions we have some rules
1- b^m × b^n = b^(m + n) ⇒ in multiplication if they have same base we add the power
2- b^m ÷ b^n = b^(m – n) ⇒ in division if they have same base we subtract the power
3- (b^m)^n = b^(mn) ⇒ if we have power over power we multiply them
4- a^m × b^m = (ab)^m ⇒ if we multiply different bases with same power then we multiply them ad put over the answer the power
5- b^(-m) = 1/(b^m) (for all nonzero real numbers b) ⇒ If we have negative power we reciprocal the base to get positive power
6- If a^m = a^n , then m = n ⇒ equal bases get equal powers
7- 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
