Exponential Functions
These are my notes and how to work on exponential functions in Algebra.
A linear function is usually written as \(g(x) = mx + b\). The variable \(m\) is the rate of change in \(g(x)\) for each unit increase in \(x\).
This all means that an exponential function is quite different than a linear function. An exponential function multiplies the previous \(y\) value by a fixed amount for each unit increase in \(x\).
For large values of \(x\), an exponential function with \(a > 1\) grows faster than any linear function.
If an exponential function is written as \(f(x) = 1a^x\) with \(a > 1\), then \(f(x)\) experiences exponential growth.
An exponential function has a variable for an exponent, whereas a polynomial function has a constant exponent. For example, \(f(x) = 3^x\) is an exponential function while \(g(x) = x^3\) is just a polynomial function.
Example 1
Simplify \(2^{-3}\)
The first thing that should catch our attention is the negative exponent. It is easier to deal with and understand if it is positive. To make it positive we move it to the denominator.
\(\frac{1}{2^3}\)
Now we just have two to the third power in the denominator.
\(\frac{1}{8}\)
Example 2
Simplify \(3(4)^.5\)
We have a decimal exponent but can see that \(.5\) is equivalent to \(\frac{1}{2}\)
A number to the one half power is the same as the square root of that same number.
The square root of four is two. Now we have:
\(3(2)\)
Our answer is six.
\(6\)
Example 3
Simplify \(-2(27)^{\frac{2}{3}}\)
Let us do the part with the exponent first
There are several ways to evaluate an exponent but I will just do it one way
If you have a preferred way do it the way you know how
\(-2(3)(3)\)
This equals:
\(-18\)
Example 4
Simplify \(\frac{1}{8}^{-1}\)
As in any problem with a negative exponent, we can just apply the reciprocal principle to \(\frac{1}{8}\) to get our exponent positive.
This gives us the answer of eight
\(8\)
Example 5
Simplify \(4^{\frac{1}{6}} 4^{\frac{1}{3}}\)
We must deal with the exponents
When we are multiplying exponents in an expression, we are adding them.
\(\frac{1}{6}+\frac{1}{3}=\frac{3}{6}=\frac{1}{2}\)
Now we have:
\(4^{\frac{1}{2}}=2\)
Example 6
Simplify \(e^x e^x\)
When we multiply exponents in an expression, just add the exponents together.
\(x+x=2x\)
This gives us:
\(e^{2x}\)
Example 7
Simply \(3^0\)
Anything to the zero power is one
\(3^0 = 1\)
Example 8
Simplify \((5^{101})^{\frac{1}{101}}\)
Combine the exponents since they are part of the same entity
This means we multiply them together
\(101 * \frac{1}{101} = 1 \)
Now we have:
\(5^{1}\)
Which just equals \(5\)