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Solve the exponential equation by rewriting the base. explain the steps towards your answer.


32^(-2x) =16

1 Answer

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Answer:

The answer is x = -0.4

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

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