Problem 78 - Elementary Algebra

Question

For exercises \(55-86\), use prime factorization to find the least common multiple. $$ 168 n^{2} w^{6} ; 252 n^{2} w^{2} $$

Step-by-Step Solution

Answer

504 n^2 w^6

Step 1 - Find prime factorization of the constants

Factorize the numerical coefficients: \[168 = 2^3 \times 3 \times 7\] \[252 = 2^2 \times 3^2 \times 7\]

Step 2 - Identify the exponents for the variables

Identify the exponents of the variables in each term: \[168 n^2 w^6\] has exponents: \[n^2, w^6\]\[252 n^2 w^2\] has exponents: \[n^2, w^2\]

Step 3 - Find the highest power of each prime

For the constants (168 and 252), take the highest powers of all primes: 2, 3, 7. Highest powers are: \[2^3 (from 168), 3^2 (from 252), 7 (common)\]

Step 4 - Find the highest power for the variables

For the variables, take the highest powers out of the given terms: \[n^2 (common), w^6 (highest from w^6)\]

Step 5 - Construct the LCM using the highest powers

Combine all the highest powers of the constants and variables: \[\text{LCM} = 2^3 \times 3^2 \times 7 \times n^2 \times w^6\]

Step 6 - Perform the multiplication

Calculate the constant part of the LCM: \[2^3 = 8, 3^2 = 9, 8 \times 9 = 72, 72 \times 7 = 504\] So, \[504 n^2 w^6\]

Key Concepts

Prime Factorization

Prime factorization breaks a number into its smallest prime number factors. For example, to factorize 168: we divide by the smallest prime, 2, to get 84. We keep dividing by 2 until we can't anymore: 84 ÷ 2 = 42, 42 ÷ 2 = 21. Next, divide by 3: 21 ÷ 3 = 7. Finally, 7 is already a prime.
So, 168 = 2^3 × 3 × 7. Similarly, for 252: 252 ÷ 2 = 126, 126 ÷ 2 = 63. Then divide by 3: 63 ÷ 3 = 21, 21 ÷ 3 = 7.
Thus, 252 = 2^2 × 3^2 × 7.
Prime factorization helps us break down large numbers into manageable parts. This makes it easier to find common multiples.

Highest Power

When finding the LCM, we need the highest power of each prime factor from the numbers we are comparing.
For 168 and 252, we have the primes 2, 3, and 7. Look at the exponents: For 2, the highest is 2^3 from 168. For 3, it is 3^2 from 252. For 7, both 168 and 252 have 7^1, so we use 7^1.
For variables, we compare their exponents too. In 168 n^2 w^6 and 252 n^2 w^2, n^2 appears in both, so we use n^2. For w, w^6 is higher than w^2.
Gathering the highest powers helps create the LCM by making sure it is divisible by both original numbers.

LCM Calculation

To find the Least Common Multiple, combine the highest powers of each prime factor and variable.
Based on our factorization, the highest powers are: 2^3, 3^2, 7, n^2, w^6.
Multiply them together: LCM = 2^3 × 3^2 × 7 × n^2 × w^6.
This ensures the LCM is divisible by both original terms: 168 n^2 w^6 and 252 n^2 w^2.
Calculate the constant part step-by-step: 2^3 = 8, 3^2 = 9, and 8 × 9 = 72, finally 72 × 7 = 504.
So, the final LCM is 504 n^2 w^6.

Exponents

Exponents represent how many times a number, or variable, is multiplied by itself.
In our problem, 168 n^2 w^6 means n is multiplied by itself 2 times and w is multiplied by itself 6 times.
This helps simplify multiplication and division of larger expressions.
When calculating the LCM, always use the highest exponent for each variable or prime factor. This ensures the LCM is large enough to contain all factors of the numbers required.