Problem 8 - Elementary Algebra

Question

For Problems \(1-30\), multiply using the properties of exponents to help with the manipulation. $$ \left(-5 x y^{2}\right)(-4 y) $$

Step-by-Step Solution

Answer

The product is \(20xy^3\).

Step 1: Identify the expression components

The expression given is \((-5xy^2)(-4y)\). It includes two terms being multiplied together: \(-5xy^2\) and \(-4y\). Each term contains numerical coefficients and variables \(x\) and \(y\) raised to certain powers.

Step 2: Multiply the numerical coefficients

Multiply the numerical coefficients of the terms: \(-5\) and \(-4\). Calculate \( -5 \times (-4) = 20.\)

Step 3: Multiply the variables

Multiply the variables separately. The term \(-5xy^2\) has variables \(xy^2\), and the term \(-4y\) has variable \(y\). 1. For \(x^1\), there is no corresponding \(x\) in the second term. So, \(x^1\) remains \(x\). 2. For \(y\), use the exponent property \(a^m \times a^n = a^{m+n}\). Add the exponents of \(y\): \(y^2 \times y^1 = y^{2+1} = y^3.\)Combine the results: \(xy^3.\)

Step 4: Write the final expression

Combine the results from the previous steps:1. Multiply the numerical coefficient from Step 2 by the variables from Step 3:\(20xy^3.\)

Key Concepts

Multiplicative Properties

Multiplicative properties are crucial in algebra, particularly when dealing with expressions that involve exponents. The main principle used in this exercise is that of multiplying coefficients and variables separately, making calculations easier and more systematic.

The critical property applied here is the rule for multiplying exponents: when the bases are the same, you can add the exponents. In mathematical terms, this property is stated as: \(a^m \times a^n = a^{m+n}\). This rule allows us to simplify expressions where the same variable is raised to different powers.

• Identify common bases between the terms being multiplied.
• Add the exponents of common bases to simplify.

These rules help in transforming complex, multi-variable equations into simpler forms that are tractable and easier to manage.

Algebraic Expressions

Understanding algebraic expressions is essential, as they are the foundation of algebra. An algebraic expression consists of numbers, variables, and operations. Variables represent unknown or changeable values, often noted by symbols like \(x\) or \(y\).

In this exercise, the expression is \((-5xy^2)(-4y)\), and we need to break it down into its components:

• \(-5xy^2\): This term includes a constant \(-5\), a variable \(x\), and \(y\) raised to the power of 2.
• \(-4y\): This term consists of a constant \(-4\) and the variable \(y\).

These components are multiplied to form a new simplified expression. Understanding these parts and how they interact through multiplication helps us make sense of complex algebraic problems and solve them more efficiently.

Step-by-Step Problem Solving

Approaching algebraic problems systematically through step-by-step problem solving can make even complex problems approachable. This methodical process involves breaking down problems into smaller, manageable parts and dealing with them in turn. Let's outline this strategy as applied to the current example.

**Step 1: Identify Components** Recognize all parts of the expression. This includes identifying coefficients and variable terms within each parenthetical unit.**Step 2: Multiply Numerical Coefficients** Focus on numerical parts of the terms first. In this case, multiply \(-5\) and \(-4\):
\(-5 \times (-4) = 20\) producing a new coefficient.**Step 3: Handle Variable Components** For variables, apply the property of exponents: combine like terms. Here, we add the exponents of the variable \(y\):
\(y^2 \times y^1 = y^{3}\).**Step 4: Formulate the Solution**Combine outcomes of both numerical and variable calculations:
The final expression derived is \(20xy^3\).Using a step-by-step approach provides clarity and comprehension, especially in handling algebraic expressions. It ensures no step is overlooked, leading to accurate results efficiently.