Problem 44 - Algebra 1

Question

Write your answer as a power or as a product of powers. $$ (-2 x y)^{3}\left(-x^{2}\right) $$

Step-by-Step Solution

Answer

\(-8x^5y^3\)

Step 1: Apply the Power of a Product Rule

The first term in our expression is \((-2 x y)^{3}\). We can use the power of a product rule here, which states that \( (a b)^n = a^n b^n \). Re-writing our expression, we get \( -2^3 x^3 y^3 \). After calculating \( -2^3 \), this gives -8, so our simplified expression is now \( -8 x^{3} y^{3} \).

Step 2: Multiply with Second Term

Then, simply multiply this result with the second term of our expression. The second term is \(-x^2\). So, our new expression becomes \( -8x^{3}y^{3} * -x^{2}\).

Step 3: Apply the Multiplication Rule for Powers

The multiplication rule states that when multiplying two expressions with the same base, we add the exponents. Here, we have \(x^3\) and \(x^2\) to multiply together. Adding the exponents, we get \( x^{3+2}\) which simplifies to \(x^5\). Our final expression then becomes \( -8x^5y^3 \).

Key Concepts

Multiplication Rule for Powers

This rule is a fundamental principle in algebra, especially when dealing with exponents. It simplifies the process of multiplying powers with the same base. The multiplication rule for powers states: when you multiply two expressions that share the same base, you simply add their exponents.
For instance, consider the multiplication of two powers like \(a^m \) and \( a^n \). The result is \(a^{m+n} \). This makes handling algebraic expressions with multiple exponents more efficient.
In our exercise, the rule was applied to the expression \(x^3\) and \(-x^2\). By adding the exponents (3 + 2), we obtained \(x^5\). This demonstrates how exponents increase linearly rather than exponentially when multiplied. It's a great way to streamline calculations in polynomials.

Simplifying Expressions

Simplifying expressions involves breaking them down into their most basic form. This process often includes using algebraic rules, like the multiplication rule for powers, and arithmetic operations to reduce complicated terms.
The initial step for simplifying our expression \((-2xy)^3\) involved using the power of a product rule. This rule helps transform the expression into a simple multiplication of separate components \(-2^3 \), \(x^3\), and \(y^3\).
After calculating \(-2^3\), we derive \(-8\), so the expression becomes \(-8x^3y^3\). Combining like terms and completing necessary arithmetic operations aids in obtaining a cleaner, more manageable expression. Thus, simplifying not only clarifies the expression's structure but also assists in finding accurate solutions easily.

Polynomial Operations

Polynomial operations encompass addition, subtraction, multiplication, and division of polynomials. They are integral to understanding high-level algebraic expressions. Multiplying polynomials requires careful attention to the placement and coefficients of terms.
In this problem, we are tasked with multiplying two polynomial expressions: \(-8x^3y^3\) and \(-x^2\). Each component in each polynomial is multiplied together, following algebraic rules.

• We begin with coefficients: multiplying \(-8\) by \(-1\), resulting in \(8\). This step is crucial as signs impact the outcome.
• The variables, \(x^3\) and \(x^2\), are multiplied using the multiplication rule for powers, giving us \(x^5\).
• The \(y^3\) remains unchanged as there's no corresponding \(y\)-term in \(-x^2\).

This operation ultimately results in a new, single polynomial: \(8x^5y^3\), illustrating how polynomials are multiplied into simplified products.