Problem 73 - Elementary Algebra

Question

For exercises \(55-86\), use prime factorization to find the least common multiple. $$ 28 x^{2} y^{5} ; 84 x y^{3} $$

Step-by-Step Solution

Answer

84 x^2 y^5

Step 1: Prime Factorization of 28

Factorize 28 into prime numbers: \[ 28 = 2^2 \times 7 \]

Step 2: Prime Factorization of 84

Factorize 84 into prime numbers: \[ 84 = 2^2 \times 3 \times 7 \]

Step 3: Write the expressions with their factorizations

Combine the factorizations with the variables: \[ 28 x^2 y^5 = 2^2 \times 7 \times x^2 \times y^5 \] and \[ 84 x y^3 = 2^2 \times 3 \times 7 \times x \times y^3 \]

Step 4: Identify the highest powers of each prime factor and variables

The highest powers of each factor are:\[ 2^2, 3, 7, x^2, y^5 \]

Step 5: Multiply the highest powers to find the LCM

Multiply the highest powers together:\[ LCM = 2^2 \times 3 \times 7 \times x^2 \times y^5 = 4 \times 3 \times 7 \times x^2 \times y^5 = 84 x^2 y^5 \]

Key Concepts

prime factorization

Prime factorization is the process of breaking down a composite number into its prime number components. Prime numbers are numbers greater than 1 that only have two divisors: 1 and themselves. For example, the prime factors of 28 are 2 and 7. To factorize 28, we breakdown it as follows:

• 28 = 2 x 14
• 14 = 2 x 7
• So, 28 = 2^2 x 7.

Repeating this process for any number gives us its unique set of prime factors. Note, this method is very useful in finding the least common multiple.

least common multiple

The least common multiple (LCM) of two or more numbers is the smallest number that is a multiple of each of the numbers. Using prime factorization helps to find the LCM easily. We start by identifying the prime factors for each number. Once we have the prime factors, we take the highest power of each prime factor from the given numbers and multiply them. For example, to find the LCM of 28 and 84 with their prime factors as:

• 28 = 2^2 x 7
• 84 = 2^2 x 3 x 7,

the LCM is computed by taking the highest powers of all prime factors present: 2^2, 3, 7. Therefore, LCM(28, 84) = 2^2 x 3 x 7 = 84.

polynomials

Polynomials are expressions consisting of variables and coefficients, combined using operations like addition, subtraction, multiplication, and non-negative integer exponents. An example is the polynomial 28x^2y^5 - 84xy^3. Here, each term is composed of a coefficient (like 28 or -84) and variables (x, y) raised to exponents. Polynomials are used to represent algebraic expressions and can be manipulated using various algebraic rules to find common multiples, roots, and solve equations.

algebraic expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operational symbols. They can be as simple as a single variable or more complex like the polynomial in our example (28x^2y^5 and 84xy^3). These expressions are integral to algebra and are used to denote quantities and relationships in equations and inequalities. By understanding how to operate with these expressions, including finding least common multiples, you can solve a wide range of algebraic problems. Always ensure that you correctly use operations and factorization methods when dealing with algebraic expressions.