Problem 98 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ (-5 x)^{3} $$

Step-by-Step Solution

Answer

\(-125x^{3}\)

Step 1: Applying Exponent Rule

First, apply the exponent to each factor inside the parentheses individually. This results in \[(-5)^{3} \cdot (x)^{3}\].

Step 2: Calculating Power of -5

Next, calculate the power of -5 which gives \[-125 \cdot x^{3}\].

Step 3: Final Simplified Expression

Lastly, since there are no like terms to combine, the simplified expression is \[-125x^{3}\].

Key Concepts

Exponent Rules

Exponent rules are a set of guidelines that specify how to handle expressions involving exponents, or powers. When you have a power such as \( a^m \), it means you multiply the base \(a\) by itself \(m\) times. In the case of \( (-5x)^3 \), we apply the exponent separately to both the coefficient and the variable.

The important rule that comes into play here is the power of a product: \( (ab)^n = a^n \cdot b^n \.\) So, for the expression given in the exercise, \( (-5x)^3 = (-5)^3 \cdot x^3 \.\) Here, we raise both -5 and \(x\) to the third power independently.

Furthermore, when dealing with negative bases, such as \( -5 \), we have to consider whether the exponent is even or odd. An even exponent will result in a positive number, whereas an odd exponent—as in this case—will give us a negative result, which is why \( (-5)^3 = -125 \).

Negative Numbers

Dealing with negative numbers is a crucial skill in algebra. A negative number is a number that is less than zero, denoted by a minus sign (–). When raising a negative number to an odd power, as in the exercise \( (-5)^3 \), the result is also negative. This is because an odd number of negative factors results in a negative product.

On the other side, if the exponent were even, the negative number would become positive because the product of an even number of negative factors is positive. For instance, \( (-5)^2 = 25 \.\)

It's also important to understand that the minus sign is actually part of the base when it's enclosed within the parentheses, and therefore, it gets raised to the power along with the base number, affecting the sign of the result.

Polynomial Expressions

A polynomial expression is an algebraic expression consisting of variables, coefficients, and the operations of addition, subtraction, multiplication, and non-negative integer exponents. The simplified expression \( -125x^3 \) is a polynomial expression, specifically, a monomial since it contains only one term.

When simplifying polynomials, it's key to combine like terms—terms that have the same variables raised to the same powers. However, in the given example, we only had one term after applying the exponent rules, so there were no like terms to combine. The exponent applies to the variable, giving us \( x^3 \.\) The combination of the coefficient \( -125 \) and the variable \( x^3 \) forms our final polynomial expression, which accurately represents the product of a constant and a variable raised to a power.