Identify the greatest common factor of $36 y^3$ and $18 y^4$.
Answer (2)
Find the greatest common factor (GCF) of the coefficients 36 and 18, which is 18.
Find the greatest common factor of the variable terms y 3 and y 4 , which is y 3 .
Multiply the GCF of the coefficients and the GCF of the variable terms: 18 × y 3 = 18 y 3 .
The greatest common factor of 36 y 3 and 18 y 4 is 18 y 3 .
Explanation
Understanding the Problem We are asked to find the greatest common factor (GCF) of 36 y 3 and 18 y 4 . The GCF is the largest expression that divides both terms without leaving a remainder.
Finding GCF of Coefficients First, let's find the GCF of the coefficients, which are 36 and 18. The greatest common factor of 36 and 18 is 18 because 18 divides both 36 and 18, and no number larger than 18 does so.
Finding GCF of Variables Next, let's find the GCF of the variable terms, which are y 3 and y 4 . The greatest common factor of y 3 and y 4 is y 3 because y 3 divides both y 3 and y 4 , and no higher power of y does so.
Combining the Results Finally, we multiply the GCF of the coefficients and the GCF of the variable terms to find the GCF of the entire expressions. So, the GCF of 36 y 3 and 18 y 4 is 18 × y 3 = 18 y 3 .
Final Answer Therefore, the greatest common factor of 36 y 3 and 18 y 4 is 18 y 3 .
Examples
Understanding the greatest common factor is useful in many real-life situations. For example, suppose you have two pieces of fabric, one is 36 inches wide and the other is 18 inches wide. You want to cut both pieces into strips of equal width, and you want the strips to be as wide as possible. The greatest common factor of 36 and 18 is 18, so you can cut both pieces of fabric into strips that are 18 inches wide. This concept extends to various scenarios like dividing resources, optimizing designs, and simplifying problems in engineering and computer science.
The greatest common factor of 36 y 3 and 18 y 4 is 18 y 3 . This is found by calculating the GCF of the coefficients, which is 18, and the GCF of the variable terms, which is y 3 . Combining these gives the final result.
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