To find the equation of an 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 follow these steps:
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Substitute the points into the equation to set up a system of equations.
For the point \( (1, \frac{1}{16}) \): \[ \frac{1}{16} = a(b)^1 \implies a(b) = \frac{1}{16} \quad \text{(1)} \]
For the point \( (3, \frac{1}{64}) \): \[ \frac{1}{64} = a(b)^3 \implies a(b^3) = \frac{1}{64} \quad \text{(2)} \]
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Express \( a \) from equation (1): \[ a = \frac{1}{16b} \quad \text{(3)} \]
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Substitute (3) into equation (2): \[ \frac{1}{64} = \left(\frac{1}{16b}\right)(b^3) \] Simplifying this gives: \[ \frac{1}{64} = \frac{b^3}{16b} \] \[ \frac{1}{64} = \frac{b^2}{16} \] Multiplying both sides by 16: \[ \frac{16}{64} = b^2 \] \[ \frac{1}{4} = b^2 \] Taking the square root: \[ b = \frac{1}{2} \quad \text{(we take the positive root since we are dealing with functions)} \]
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Substitute \( b \) back into equation (3) to find \( a \): \[ a = \frac{1}{16 \cdot \frac{1}{2}} = \frac{1}{8} \]
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Write the final equation using the values of \( a \) and \( b \): \[ y = \frac{1}{8} \left(\frac{1}{2}\right)^x \]
Thus, the equation for the exponential function is: \[ y = \frac{1}{8} \left(\frac{1}{2}\right)^x \]