Problem 86 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ -4 x y^{2}\left[7 x y-6\left(5-x y^{2}\right)+3(-x y+1)+1\right] $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression $$-4 x y^{2}\left[ 7 x y - 6\left(5-x y^{2}\right) + 3(-x y + 1) + 1\right]$$. Answer: After simplifying the algebraic expression, we get $$-56xy^3 + 136xy^2 - 24x^2y^4$$.

Step 1: Distribute the given expression

First, distribute the term \(-4xy^2\) into each term inside the brackets. We have: $$ -4 x y^{2}\left[ 7 x y - 6\left(5-x y^{2}\right) + 3(-x y + 1) + 1\right] $$

Step 2: Apply the distributive property

Now, we apply the distributive property and simplify each term: $$ -28x^2y^3 + 24x y^2 (5- x y^2) - 12x y^3 + 4x y^2 + 3(-4x y^3 + 4x y^2) $$

Step 3: Combine like terms

Next, we combine the like terms: $$ -28x^2y^3 - 12x y^3 + 120x y^2 - 24x^2 y^4 - 4x y^3 + 4x y^2 -12x y^3 + 12x y^2 $$

Step 4: Rearrange and simplify

Now we rearrange the terms and simplify the expression: $$ (-28x^2y^3 - 12x y^3 - 4x y^3 -12x y^3) + (120x y^2 + 4x y^2 + 12x y^2) - 24x^2 y^4 $$ $$ -56x y^3 + 136x y^2 - 24x^2 y^4 $$

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps simplify complex expressions. Essentially, it allows you to multiply a single term by each term within a set of parentheses. Applying this property ensures that you distribute multiplication over addition or subtraction within the brackets.
For example, if you have an expression like \(a(b + c)\), the distributive property lets you rewrite it as \(ab + ac\). This not only makes calculations easier, but also clarifies the expression by removing the parentheses.
In the original problem, the term \(-4xy^2\) is multiplied across several terms confined within square brackets. This requires distributing \(-4xy^2\) to every individual term inside the brackets, enabling further simplification. Remember to keep signs in mind when distributing—multiplying negative terms changes the sign of the resulting terms. These steps are crucial as they lay the foundation for the subsequent simplification process.

Combining Like Terms

Combining like terms is a method used in algebra to simplify expressions by merging terms that have the same variable part. When terms share the same variable raised to the same power, they can be combined by adding or subtracting their coefficients.
Consider an expression like \(3xy + 5xy\). Here, \(3xy\) and \(5xy\) are like terms because they share the same variable, \(xy\). They combine to form a single term: \((3 + 5)xy = 8xy\).
This process of combining like terms is a simplification technique that makes expressions more manageable and reduces the number of terms to deal with. In the provided solution, we see terms like \(-12xy^3 - 4xy^3 -12xy^3\) combined to make \(-56xy^3\). Remember, organizing and grouping similar terms is essential for efficient simplification. Be attentive to ensure terms combine correctly based on their variables and exponents.

Algebraic Simplification

Algebraic simplification is the art of making algebraic expressions more compact and easier to work with. It's about reducing expressions to their simplest form while retaining their original meaning.
Simplification can be achieved through several processes, including applying the distributive property, combining like terms, and arranging terms in an expression logically. This involves meticulously reviewing each step to ensure no mistakes are made, and each part of the expression is correctly simplified.
Going through the example in the original exercise, the final expression resulted in \(-56xy^3 + 136xy^2 - 24x^2y^4\). Simplification helped distill a complex expression to this form. This makes it easier to interpret and work with, especially when dealing with subsequent algebraic operations. Remember, a thoroughly simplified expression enhances comprehensibility and minimizes potential errors in further calculations.