b) [tex]4^{-x} = \frac{1}{16}[/tex]
c) [tex]\frac{1}{5^n} = 125[/tex]
Answer (1)
Let's solve each equation one by one.
b) 4 − x = 16 1
To solve this equation, we need to write both sides of the equation with the same base. Notice that 16 = 4 2 . Also, 4 2 1 = 4 − 2 .
So, we can rewrite the equation as:
4 − x = 4 − 2
Since the bases are the same, we can set the exponents equal to each other:
− x = − 2
Solving for x , we multiply both sides by − 1 :
x = 2
So, the solution is x = 2 .
c) 5 n 1 = 125
First, recognize that 125 is a power of 5 :
125 = 5 3
Rewriting 5 n 1 as a negative exponent gives us:
5 − n = 5 3
With identical bases, we equate the exponents:
− n = 3
Solving for n involves multiplying by − 1 :
n = − 3
Thus, the solution is n = − 3 .
Both problems involve recognizing powers and relating them using properties of exponents. It is important to express numbers in terms of their base powers to easily equate the exponents.
