Problem 12 - Algebra 1: Concepts and Skills

Question

Simplify the expression. \(6(-x)^{3}\)

Step-by-Step Solution

Answer

The simplified expression for \(6(-x)^{3}\) is \(-6x^{3}\).

Step 1: Application of Exponent Rule

When a negative number is raised to an odd power, the result is negative. Hence, \((-x)^{3} = -x^{3}\). This happens because the cube of a number includes three multiplications of that number. In this case, it means multiplying -x three times. The negative sign will retain as the power is odd.

Step 2: Multiplying Constant and Variable

Now, we need to multiply 6 with \(-x^{3}\). Following the basic rules of multiplication in algebra: a number times a negative variable equals a negative number times the variable. So, it simplifies to \(6 * -x^{3} = -6x^{3}\).

Key Concepts

Negative Exponents

When dealing with exponents, particularly negative exponents, it can feel tricky at first. But, once you understand the rule, it becomes much simpler. A negative exponent means that the base is on the wrong side of the fraction—so, to "fix" it, you move its position. For example, if you have a base with a negative exponent like

• \((-x)^{-3}\)

this means you should think of it as:

• \(\frac{1}{(-x)^{3}}\)

This rule helps when simplifying expressions because it tells where the base should "live"—in the numerator or the denominator. Remember, the negative sign here only indicates the position change, not that the base itself is negative.
This concept is different from applying odd powers, where the sign affects the overall result, as we’ll see in the next section.

Odd Powers

Odd powers have a unique property. When a base (like a number or variable) with a negative sign is raised to an odd power, the entire expression retains a negative sign. This is different from even powers, which turn negatives into positives. Here's why:

• Consider the expression \((-x)^{3}\). The exponent 3 is odd.
• If you expand this, it looks like \(-x \times -x \times -x\).
• The negatives in two of those multiplies will cancel each other out, leaving a single negative from \(-x\).

Thus, the expression simplifies to \(-x^{3}\), which retains the negative because of the single uncancelled negative from multiplying three times.
This feature of odd powers is crucial when simplifying expressions, as it directly influences the overall sign of your result.

Multiplying Constants with Variables

Multiplying constants with variables is a fundamental algebraic operation. When you have a numeric constant (like 6) and you're multiplying it by a variable expression (like \(-x^{3}\)), follow these straightforward steps:

• First, consider the factors separately. The constant in our example is 6.
• Next, look at the variable part, \(-x^{3}\).
• Apply the rules of multiplication: multiplying a positive constant with a negative variable expression results in a negative product.

In this case, multiplying 6 by \(-x^{3}\) results in \(-6x^{3}\). Essentially, you're distributing the constant and maintaining the negative sign throughout the expression.
This simple rule helps when combining terms correctly, ensuring that the final simplified expression correctly reflects all signs and numerical factors.