Math  /  Algebra

QuestionFind the greatest common factor of these two expressions. 18v8y7u3 and 27y5u418 v^{8} y^{7} u^{3} \text { and } 27 y^{5} u^{4}

Studdy Solution

STEP 1

What is this asking? We need to find the biggest expression that divides evenly into both 18v8y7u318 v^{8} y^{7} u^{3} and 27y5u427 y^{5} u^{4}. Watch out! Don't forget to consider *all* the variables and their *lowest* powers!

STEP 2

1. Find the Greatest Common Factor (GCF) of the coefficients.
2. Find the GCF of each variable.
3. Combine to get the final GCF.

STEP 3

Let's **find the GCF** of the coefficients, which are the numbers in front of the variables.
Our coefficients are **18** and **27**.
What's the biggest number that divides evenly into both?

STEP 4

We can list out the factors of each.
Factors of **18** are 1, 2, 3, 6, 9, and 18.
Factors of **27** are 1, 3, 9, and 27.
The **largest factor they share** is **9**!

STEP 5

Now, let's look at the variables.
We have *v*, *y*, and *u*.
Remember, the GCF uses the *lowest* power of each variable that appears in *both* expressions.

STEP 6

The variable *v* only appears in the first expression, 18v8y7u318 v^{8} y^{7} u^{3}, so it *won't* be part of our GCF.

STEP 7

The variable *y* appears in both expressions.
The lowest power is y5y^{5}, so that's what we'll use.

STEP 8

The variable *u* also appears in both expressions.
The lowest power is u3u^{3}, so we'll use that.

STEP 9

Now, we **put it all together**!
We multiply the GCF of the coefficients and the GCF of each variable.

STEP 10

Our GCF of the coefficients was **9**.
Our GCF for the variables is y5u3y^{5} \cdot u^{3}.

STEP 11

Multiplying them together gives us the **final GCF**: 9y5u39 y^{5} u^{3}.
Boom!

STEP 12

The greatest common factor of 18v8y7u318 v^{8} y^{7} u^{3} and 27y5u427 y^{5} u^{4} is 9y5u39 y^{5} u^{3}.

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