How to find the greatest common factor with exponents?
Understand the Problem
The question is asking for a method to determine the greatest common factor (GCF) of numbers that are expressed with exponents. This involves identifying the common bases and taking the lowest exponent for each base.
Answer
The GCF of $a^m$ and $b^n$ is the product of common bases raised to their lowest exponents. For example, for $2^3$ and $2^5$, it is $8$.
Answer for screen readers
The final answer for the GCF of the numbers expressed with exponents is the product of each common base raised to the lowest exponent.
For instance, if we find the GCF of $2^3$ and $2^5$, it would be $2^3 = 8$.
Steps to Solve
- Identify the Numbers and Their Prime Factorization
List out the numbers for which you need to find the GCF and perform the prime factorization.
For example, let's say we need the GCF of $a^m$ and $b^n$.
- Extract Common Bases
Identify the bases that are common between the two numbers. Let’s say they are both expressed as prime factors.
For instance, if you have $2^3$ and $2^5$, the common base is $2$.
- Determine the Lowest Exponent of Common Bases
For each common base found in the previous step, take the lowest exponent.
Using the example from step 2: The lowest exponent for the base $2$ in our example is $3$ from $2^3$.
- Construct the GCF Expression
Now, express the GCF using the common bases and their lowest exponents.
In the previous example, the GCF would be expressed as $2^3$.
- Calculate the GCF
If needed, calculate the actual numerical value of the GCF by evaluating the expression found in the previous step.
For our example, $2^3 = 8$, so the GCF of $2^3$ and $2^5$ is $8$.
The final answer for the GCF of the numbers expressed with exponents is the product of each common base raised to the lowest exponent.
For instance, if we find the GCF of $2^3$ and $2^5$, it would be $2^3 = 8$.
More Information
The greatest common factor (GCF) is useful in simplifying fractions, factoring polynomials, and finding common denominators in fraction problems. Understanding how to deal with numbers expressed in exponents can help in more advanced algebra and number theory.
Tips
- Forgetting to look for common bases: Always make sure to check if both numbers share the same prime factors.
- Confusing exponents: When identifying the lowest exponent, it's important not to assume the largest is the correct one; just be sure to pick the smallest.
