Problem 48 - Algebra A Combined Function

Question

Perform each indicated operation. $$ \left(-5 x^{2}\right)\left(x^{2}\right) $$

Step-by-Step Solution

Answer

The result is \(-5x^4\).

Step 1: Review the Expression

We need to multiply two algebraic expressions: \(-5x^2\) and \(x^2\). This means we are performing multiplication on coefficients and adding exponents for the same base, \(x\).

Step 2: Multiply the Coefficients

Identify the coefficients of the terms. In this case, \(-5x^2\) has a coefficient of \(-5\) and \(x^2\) has an implicit coefficient of \(1\). Multiply these coefficients: \(-5 \times 1 = -5\).

Step 3: Apply the Power Rule to the Variables

Next, apply the rule for multiplying powers: when you multiply two powers with the same base, you add the exponents. The expression has \(x^2 \times x^2\), so add the exponents: \(2 + 2 = 4\).

Step 4: Formulate the Result

Combine the results from prior steps to create the simplified expression. The coefficient is \(-5\) and the new power of \(x\) is \(x^4\). So, \(-5x^2 \times x^2 = -5x^4\).

Key Concepts

Understanding Coefficients

In algebra, coefficients are the numbers that appear before the variables in an expression. These numbers multiply the variable they are associated with. Coefficients can be viewed as the scaling factor for the term. For example, in the expression \(-5x^2\), the coefficient is \(-5\). This tells you that the term is scaled by a factor of \(-5\).
Whenever you are multiplying algebraic expressions, you should first multiply the coefficients across each term involved in the operation. In our example, when multiplying \(-5x^2\) and \(x^2\), our coefficients are \(-5\) and \(1\) (since \(x^2\) is equal to \(1 \cdot x^2\)). So, you multiply these coefficients:

• Step: \(-5 \times 1 = -5\)

This outcome sets the stage for the final coefficient of the simplified expression.

Exploring Powers and Exponents

Powers and exponents are a way to represent repeated multiplication succinctly. The exponent indicates how many times you multiply the base by itself. In the expression \(x^2\), \(2\) is the exponent, and \(x\) is the base. It tells you that \(x\) should be multiplied by itself twice, or \(x \cdot x\).
When multiplying terms with the same base, you can add their exponents. This is a convenient rule because it allows for simplification. For example, in multiplying \(x^2\) by \(x^2\), you will add the exponents:

• Step: \(2 + 2 = 4\)

This rule helps in transforming the expression \(x^2 \times x^2\) into \(x^4\), simplifying it considerably.

Simplification of Expressions

Simplification is the process of taking an expression and rewriting it in its most basic form, which often involves performing operations with coefficients and powers.
After multiplying coefficients and handling powers, you combine these results to form a single term. The expression \(-5x^2 \times x^2\) reduces down to:

• Coefficients: \(-5\)
• Exponent of \(x\): \(4\)

Thus, bringing these together, we formulate the final simplified expression:

\(-5x^4\)

Simplification makes it easier to visualize and work with algebraic expressions, providing clarity and accuracy for further calculations or evaluations.