Problem 55 - Algebra 1

Question

Simplify the variable expression. $$3(-x)(-x)(-x)$$

Step-by-Step Solution

Answer

\(-3x^3\)

Step 1: Apply Distributive Property

First, distribute the 3 to each of the \(-x\), giving you \(-3x * -3x * -3x\).

Step 2: Simplify Expressions

Now, \(-x * -x * -x\) simplifies to \(-x^3\) since a negative times a negative gives a positive and then multiplied by a negative again gives a negative. Therefore, the expression simplifies to \(-3x^3\).

Key Concepts

Distributive Property

The distributive property is a key concept in algebra that lets us simplify expressions by distributing one term across the others within parentheses. Essentially, it means multiplying the term outside the parenthesis with each term inside the parenthesis. For example, if you have an expression like \(a(b + c)\), you would apply the distributive property by calculating \(ab + ac\). It's a straightforward but powerful property that helps break down complex expressions into simpler parts.
In the original exercise, we start with the expression \(3(-x)(-x)(-x)\). Though there's no addition or subtraction inside the parentheses here, we treat each \(-x\) as a separate entity that involves multiplication. Therefore, the distributive property guides us to distribute the 3 to each \(-x\), which means multiplying each instance of \(-x\) by 3. This might look subtly different, but it's an application of the same fundamental principle of distributing multiplication evenly.

Negative Signs in Multiplication

When dealing with negative signs in multiplication, the rules are straightforward. Knowing how negatives interact in arithmetic is crucial for simplifying expressions correctly. Here are some essential rules to remember:

• The product of two negative numbers is positive. For example, \(-a \times -b = ab\).
• If you multiply a positive number by a negative number, the result is negative. For example, \(a \times -b = -ab\).
• The product of three negatives is negative again. For instance, \(-a \times -b \times -c = -abc\).

In our expression \(3(-x)(-x)(-x)\), applying these rules step by step is crucial. First, \(-x\) multiplied by \(-x\) results in \(x^2\), a positive term. Then, multiplying this positive result by another \(-x\) gives us \(-x^3\). Finally, multiplying by the leading 3, we end up with \(-3x^3\). Understanding these simple rules helps avoid common mistakes when simplifying expressions with negative signs.

Exponentiation

Exponentiation is a mathematical operation that denotes repeated multiplication of a number by itself. When you see an expression like \(x^3\), it means \(x\times x\times x\). Exponents denote how many times the base (in this case, \(x\)) appears in the multiplication. Here are some key points to remember about exponentiation:

• An exponent of 1 means the number appears once, so \(x^1 = x\).
• An exponent of 0 equals 1, meaning \(x^0 = 1\), no matter what the base is, as long as it's not zero.
• An exponent of 2 is called 'squared' (\(x^2\)), while an exponent of 3 is called 'cubed' (\(x^3\)).

So, in our simplification problem, when \(-x\) is multiplied by itself twice \((-x)(-x)\), we use the exponent 3 to represent this operation, yielding \(x^3\). Finally, with the negative sign from \(-x\), the whole expression becomes \(-x^3\). Learning exponentiation helps you manage and simplify expressions with repeated multiplication like a pro.