Question

Simplify the variable expression. $$(-4)(-c)(-c)(-c)$$

Short answer

The simplified form of the expression is \(-4c^3\).

Step-by-step solution

Multiply the coefficient

First, multiply the constant coefficients together, using the rule that the product of two negative numbers is positive, and the product of an even number of negative numbers is positive. So, \(-4 \times -1 \times -1 \times -1 = -4\).

State the variable

Since all variables are the same and they are multiplied together, you keep the variable which is 'c' and write it three times because it is repeated three times in the multiplication.

Compile the Result

Combine the result from the multiplication of coefficients and the variable. The result is \(-4c^3\), which means -4 times c cubed.

Key concepts

Multiplying Coefficients

Simplifying variable expressions often starts with dealing with coefficients. The term 'coefficient' refers to a numerical or constant factor in an algebraic term that is multiplied by the variable. For instance, in the expression \(3x\), 3 is the coefficient of the variable \(x\).

When you're multiplying coefficients, the process is similar to multiplying any other numbers. But keep in mind a few basic rules of multiplication:

• A positive number multiplied by a positive number will give you a positive result.
• A negative number multiplied by a positive number will give you a negative result.
• A negative number multiplied by a negative number will give you a positive result.

The last point is particularly important in our original exercise, where the coefficient \( -4 \) is a negative number. The rule tells us that an even number of negative coefficients will result in a positive product. This is why in our exercise, multiplying the negative coefficients resulted in a negative outcome, as there were three negative factors along with the negative coefficient, resulting in an odd number of negatives.

Negative Numbers Multiplication

When working with negative numbers, it's crucial to understand how they interact during multiplication. A negative number can be perceived as the opposite of a positive number, and when multiplying them together, the rules are straightforward: two negatives make a positive, and a positive and a negative make a negative.

Consider this practical approach: imagine negative numbers as directions. If two negative numbers (two left turns, for instance) are multiplied, they bring you back in the positive direction (right turn). This concept is applied in our exercise where multiplying the negative coefficient \( -4 \) by three instances of \( -c \) brought about a negative result \( -4c^3 \), as multiplying four negatives results in a positive number. Remembering this rule can help prevent errors when simplifying variable expressions involving negative numbers.

Variables in Algebra

Variables are symbols, often letters, that represent numbers in algebraic expressions. They are crucial because they can stand for unknown values that we can solve for, or they may represent values that can change within a given problem. When dealing with variables, here are some useful tips:

• The same variable is treated as the same unknown value throughout an equation or expression.
• When they are multiplied together, we use the laws of exponents to combine them.
• Always combine like terms — that is, terms with the same variables raised to the same power — to simplify expressions.

In the provided exercise, the variable \(c\) is repeated three times, which means \(c\) is being multiplied by itself three times. In such a scenario, we use exponents to simplify the expression, leading to \(c^3\) in the final answer.

Exponents

Exponents are a convenient way to represent repeated multiplication of the same number or variable. The exponent is the small number placed above and to the right of the base number and signifies how many times the base number is used as a factor.

Let's break down some key points about exponents:

• An exponent of 2 (also called a square) means the base is multiplied by itself once: \(a^2 = a \times a\).
• An exponent of 3 (also called a cube) uses the base as a factor three times: \(a^3 = a \times a \times a\).
• Any base except zero, raised to the power of zero, equals one (\(a^0 = 1\), where \(a \eq 0\)).

Therefore, in our provided exercise \( (-c)(-c)(-c) \), the variable \(c\) is used as a factor three times, which is why we simplify it using the exponent 3, resulting in \(c^3\). Multiplying it by the coefficient from the previous steps gives us the simplified result \( -4c^3 \). Understanding exponents helps greatly in simplifying expressions and determining the degree of polynomials in more advanced algebra.