Question
An exponential function of form f left parenthesis x right parenthesis equals b Superscript xf(x)=bx includes the points (2, 16), (3, 64), and (4, 256). What is the value of b?
Answers
GPT-4o mini
To find the value of \( b \) in the exponential function \( f(x) = b^x \) given the points \( (2, 16) \), \( (3, 64) \), and \( (4, 256) \), we can use these points to create equations.
1. Using the point \( (2, 16) \):
\[
f(2) = b^2 = 16
\]
2. Using the point \( (3, 64) \):
\[
f(3) = b^3 = 64
\]
3. Using the point \( (4, 256) \):
\[
f(4) = b^4 = 256
\]
Now, we can solve for \( b \) from the first equation:
\[
b^2 = 16 \implies b = \sqrt{16} = 4.
\]
Next, we can check the second equation:
\[
b^3 = 64 \implies 4^3 = 64 \quad \text{(which is true)}.
\]
Finally, checking the third equation:
\[
b^4 = 256 \implies 4^4 = 256 \quad \text{(which is also true)}.
\]
Thus, \( b \) is consistently found to be 4 across all points.
Therefore, the value of \( b \) is
\[
\boxed{4}.
\]
1. Using the point \( (2, 16) \):
\[
f(2) = b^2 = 16
\]
2. Using the point \( (3, 64) \):
\[
f(3) = b^3 = 64
\]
3. Using the point \( (4, 256) \):
\[
f(4) = b^4 = 256
\]
Now, we can solve for \( b \) from the first equation:
\[
b^2 = 16 \implies b = \sqrt{16} = 4.
\]
Next, we can check the second equation:
\[
b^3 = 64 \implies 4^3 = 64 \quad \text{(which is true)}.
\]
Finally, checking the third equation:
\[
b^4 = 256 \implies 4^4 = 256 \quad \text{(which is also true)}.
\]
Thus, \( b \) is consistently found to be 4 across all points.
Therefore, the value of \( b \) is
\[
\boxed{4}.
\]