QuestionFind the least common multiple of $16 u^{3} x^{4}$ and $6 u^{7} v^{5} x^{2}$ .

Studdy Solution

STEP 1

Assumptions1. We are asked to find the least common multiple (LCM) of two algebraic expressions. . The two expressions are $16u^{3}x^{4}$ and $6u^{7}v^{5}x^{}$ .
3. The LCM of two algebraic expressions is the smallest expression that is a multiple of both expressions.

STEP 2

We start by finding the LCM of the numerical coefficients, which are16 and6 in this case. The LCM of two numbers is the smallest number that is a multiple of both numbers.
$LCM(16,6)$

STEP 3

To find the LCM of16 and6, we can list the multiples of each number until we find a common multiple.
Multiples of1616,32,48,64,80,96,112,128,144,160, ...
Multiples of66,12,18,24,30,36,42,48, ...
So, the LCM of16 and6 is48.

STEP 4

Next, we find the LCM of the variables. For each variable, we take the highest power that appears in either expression.
For $u$ , the powers are3 and7. So, the highest power is7. Therefore, $u^{7}$ is part of the LCM.
For $v$ , the power is in the second expression and it doesn't appear in the first expression. So, $v^{}$ is part of the LCM.
For $x$ , the powers are4 and2. So, the highest power is4. Therefore, $x^{4}$ is part of the LCM.

STEP 5

Now, we multiply the LCM of the numerical coefficients and the LCM of the variables to get the LCM of the two expressions.
$LCM =48u^{7}v^{5}x^{4}$ The least common multiple of the two expressions $16u^{3}x^{4}$ and $u^{7}v^{5}x^{2}$ is $48u^{7}v^{5}x^{4}$ .

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