Exponential Functions Cont.
Evaluating Exponential Functions
To evaluate an exponential function with the form , we simply substitute with the given value, and calculate the resulting power.
Let . What is ?
To evaluate an exponential function with a form other than the basic form, it is important to follow the order of operations.
Let . What is ?
Note that if the order of operations were not followed, the result would be incorrect:
How To: Evaluating Exponential Functions
- Given an exponential function, identify , , and the value of you’re being asked to substitute into the function.
- Replace the variable in the function with the given number.
- Compute the value of . This means raising the base to the power of .
- If there is a coefficient in front of the base, multiply the result of by . If is , this step does not change the value.
- Simplify the expression if necessary. This could involve performing any additional multiplication or addition/subtraction if the function has more terms.
Let . Evaluate without using a calculator.
Suppose a particular population of bacteria is known to double in size every hours. If a culture starts with bacteria, the number of bacteria after hours is . The number of bacteria after hours is .
In general, the number of bacteria after hours is . Letting , we see that the number of bacteria after hours is .
Find the number of bacteria after hours, hours, and hours.
Laws of Exponents
The Laws of Exponents are fundamental rules that govern the operations involving powers. These rules are essential for simplifying expressions and are foundational for higher-level math.
laws of exponents
- The Product of Powers rule states that when you multiply two exponents with the same base, you can add the exponents.
- The Quotient of Powers rule tells us that when dividing exponents with the same base, we subtract the exponents.
- The Power of a Power rule shows that when taking an exponent to another exponent, we multiply the exponents.
- The Power of a Product rule lets us know that when raising a product to an exponent, each factor in the product is raised to the exponent.
- The Power of a Quotient rule indicates that when a quotient is raised to an exponent, both the numerator and the denominator are raised to the exponent.
Note: This is true for any constants , and for all and
Use the laws of exponents to simplify each of the following expressions.
When you encounter a negative exponent on a term in the denominator of a fraction, you can transform it into a positive exponent by moving the term to the numerator.
Using this rule can significantly simplify expressions involving exponents.