To find the equation of the exponential function in the form \( y = a(b)^x \) that passes through the points \( (1, \frac{1}{16}) \) and \( (3, \frac{1}{64}) \), we will use these points to set up a system of equations.
From the first point \( (1, \frac{1}{16}) \): \[ \frac{1}{16} = a(b)^1 \quad \Rightarrow \quad ab = \frac{1}{16} \quad \text{(1)} \]
From the second point \( (3, \frac{1}{64}) \): \[ \frac{1}{64} = a(b)^3 \quad \Rightarrow \quad ab^3 = \frac{1}{64} \quad \text{(2)} \]
Now we have the system of equations:
- \( ab = \frac{1}{16} \)
- \( ab^3 = \frac{1}{64} \)
We can substitute equation (1) into equation (2). From equation (1), we have: \[ a = \frac{1}{16b} \]
Substituting \( a \) into equation (2): \[ \left(\frac{1}{16b}\right)b^3 = \frac{1}{64} \] \[ \frac{b^2}{16} = \frac{1}{64} \]
Now, multiply both sides by 16: \[ b^2 = \frac{16}{64} = \frac{1}{4} \]
Taking the square root of both sides gives us: \[ b = \frac{1}{2} \quad \text{(since b must be positive for growth)} \]
Now, substitute \( b = \frac{1}{2} \) back into equation (1) to find \( a \): \[ a \left(\frac{1}{2}\right) = \frac{1}{16} \] \[ a = \frac{1}{16} \cdot 2 = \frac{1}{8} \]
Now that we have determined \( a \) and \( b \), the equation of the exponential function is: \[ y = \frac{1}{8} \left(\frac{1}{2}\right)^x \]
Alternatively, you can write it in the form: \[ y = \frac{1}{8} \cdot 2^{-x} \]
Thus, the final equation is: \[ \boxed{y = \frac{1}{8} \left(\frac{1}{2}\right)^x} \]
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