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Answer :
To find the least common multiple (LCM) of the expressions [tex]\(10x\)[/tex] and [tex]\(45x^4\)[/tex], we'll break it down into two parts: the numerical coefficients and the variables.
### Step 1: Find the LCM of the Numerical Coefficients
First, consider the numerical parts of the expressions, which are 10 and 45. To find the LCM of these numbers:
1. Prime Factorization:
- [tex]\(10 = 2 \times 5\)[/tex]
- [tex]\(45 = 3^2 \times 5\)[/tex]
2. Identify Common and Unique Factors:
- Here, the factors are: [tex]\(2, 3^2, 5\)[/tex].
3. Take the Highest Power of Each Prime Factor:
- [tex]\(2^1\)[/tex] from 10 (since 2 only appears in 10),
- [tex]\(3^2\)[/tex] from 45,
- [tex]\(5^1\)[/tex] from both (it's the same in both, so we take just one).
4. Calculate the LCM for the Coefficients:
[tex]\[
\text{LCM} = 2 \times 3^2 \times 5 = 2 \times 9 \times 5 = 90
\][/tex]
### Step 2: Combine with the Variable Part
Now consider the variable parts, [tex]\(x\)[/tex] and [tex]\(x^4\)[/tex]. We need to find the LCM in terms of the variable [tex]\(x\)[/tex]:
1. Identify the Highest Power of the Variable:
- In [tex]\(10x\)[/tex], the power of [tex]\(x\)[/tex] is [tex]\(1\)[/tex].
- In [tex]\(45x^4\)[/tex], the power of [tex]\(x\)[/tex] is [tex]\(4\)[/tex].
- The maximum power is [tex]\(x^4\)[/tex].
### Step 3: Combine the Results
Combine the LCM of the coefficients with the highest power of the variable:
[tex]\[
\text{LCM of } 10x \text{ and } 45x^4 = 90x^4
\][/tex]
Thus, the least common multiple of [tex]\(10x\)[/tex] and [tex]\(45x^4\)[/tex] is option (a) [tex]\(90x^4\)[/tex].
### Step 1: Find the LCM of the Numerical Coefficients
First, consider the numerical parts of the expressions, which are 10 and 45. To find the LCM of these numbers:
1. Prime Factorization:
- [tex]\(10 = 2 \times 5\)[/tex]
- [tex]\(45 = 3^2 \times 5\)[/tex]
2. Identify Common and Unique Factors:
- Here, the factors are: [tex]\(2, 3^2, 5\)[/tex].
3. Take the Highest Power of Each Prime Factor:
- [tex]\(2^1\)[/tex] from 10 (since 2 only appears in 10),
- [tex]\(3^2\)[/tex] from 45,
- [tex]\(5^1\)[/tex] from both (it's the same in both, so we take just one).
4. Calculate the LCM for the Coefficients:
[tex]\[
\text{LCM} = 2 \times 3^2 \times 5 = 2 \times 9 \times 5 = 90
\][/tex]
### Step 2: Combine with the Variable Part
Now consider the variable parts, [tex]\(x\)[/tex] and [tex]\(x^4\)[/tex]. We need to find the LCM in terms of the variable [tex]\(x\)[/tex]:
1. Identify the Highest Power of the Variable:
- In [tex]\(10x\)[/tex], the power of [tex]\(x\)[/tex] is [tex]\(1\)[/tex].
- In [tex]\(45x^4\)[/tex], the power of [tex]\(x\)[/tex] is [tex]\(4\)[/tex].
- The maximum power is [tex]\(x^4\)[/tex].
### Step 3: Combine the Results
Combine the LCM of the coefficients with the highest power of the variable:
[tex]\[
\text{LCM of } 10x \text{ and } 45x^4 = 90x^4
\][/tex]
Thus, the least common multiple of [tex]\(10x\)[/tex] and [tex]\(45x^4\)[/tex] is option (a) [tex]\(90x^4\)[/tex].
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