Problem 108 - Intermediate Algebra

Question

Perform the indicated operations. $$ \left(5 k-6 m^{2}\right)^{2} $$

Step-by-Step Solution

Answer

The expanded form is \(25k^2 - 60km^2 + 36m^4\).

Step 1: Recognize the Formula Needed

This exercise involves squaring a binomial expression, namely \((5k - 6m^2)^2\). Recognize that the formula for the square of a binomial \((a - b)^2\) is given by: \(a^2 - 2ab + b^2\).

Step 2: Identify Components

Identify the components of the binomial: here, \(a = 5k\) and \(b = 6m^2\). This setup will help in applying the formula correctly.

Step 3: Apply the Binomial Square Formula

Substitute \(a = 5k\) and \(b = 6m^2\) into the formula \(a^2 - 2ab + b^2\): \((5k)^2 - 2(5k)(6m^2) + (6m^2)^2\).

Step 4: Simplify Each Term

Calculate each of the individual terms: - \((5k)^2 = 25k^2\)- \(-2(5k)(6m^2) = -60km^2\)- \((6m^2)^2 = 36m^4\).

Step 5: Combine the Terms for Final Result

Combine the simplified terms from Step 4 to get the final expanded expression: \(25k^2 - 60km^2 + 36m^4\).

Key Concepts

Square of a Binomial

Squaring a binomial involves expanding an expression of the form \((a - b)^2\). Many students come across this in their early algebra lessons, and it's a crucial skill for understanding more complex algebraic manipulations. The trick is to use the formula:

• a² is the square of the first term,
• -2ab represents twice the product of the first and second term,
• b² is the square of the second term.

In our example, the binomial is \((5k - 6m^2)^2\). It means:

• First, take the square of the term \(5k\) to get \(25k^2\).
• Next, double the product of \(5k\) and \(6m^2\) to get \(-60km^2\).
• Finally, square \(6m^2\) to obtain \(36m^4\).

Understanding this formula makes it much quicker to expand binomials, allowing you to solve these kinds of problems efficiently.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations (like addition or subtraction). These expressions can be simple, like \(2x + 3\), or more complex, like \(5k^2 - 60km^2 + 36m^4\). Learning how to manipulate these expressions is a foundational algebra skill. Let’s look into the expression \((5k - 6m^2)^2\). This is an example of a polynomial expression:

• It contains variables \(k\) and \(m\), which can hold any value.
• We apply arithmetic operations, such as multiplication and addition, to these variables.
• The degree of a polynomial is determined by the highest power of any variable in the expression.

In our expanded result \(25k^2 - 60km^2 + 36m^4\), the highest degree of a term is 4, found in \(36m^4\). Knowing this helps understand the behavior and characteristics of the expression and prepares you for solving algebraic equations effectively.

Simplification of Expressions

Simplifying an algebraic expression involves rewriting it in its simplest form. The goal is to reduce complexity while maintaining the expression's original value. This is done by combining like terms and using arithmetic operations efficiently. Consider the expression \(25k^2 - 60km^2 + 36m^4\), derived from the expansion of the binomial \((5k - 6m^2)^2\). Here’s how simplification works step by step:

• Identify and group like terms. In this case, there are no like terms to combine since each term is distinct by its variable parts.
• Ensure that each term is expressed correctly and simplify each as much as possible. This involves checking calculations involving coefficients and exponents.

Thus, \(25k^2 - 60km^2 + 36m^4\) is the simplified version of our original problem. Simplifying effectively can make complex problems more manageable, aiding in both understanding and further calculations.