Problem 22 - Algebra 2

Question

Simplify. Assume that no variable equals 0. $$ (-2 c)^{3} $$

Step-by-Step Solution

Answer

(-2c)^{3} = -8c^{3}.

Step 1: Understand the Expression

The expression given is (-2c)^{3}. This means we need to cube the entire expression inside the parentheses, which includes the coefficient -2 and the variable c.

Step 2: Apply the Power to the Coefficient

When cubing the coefficient, (-2), we calculate (-2)^{3}. This means multiplying -2 by itself three times: (-2) imes (-2) imes (-2).

Step 3: Calculate the Cubed Coefficient

Compute the value: (-2) imes (-2) = 4, and then multiply 4 by -2 which gives us -8. So, (-2)^{3} = -8.

Step 4: Apply the Power to the Variable

Now, cube the variable c by calculating c^{3}, which means multiplying c by itself three times: c imes c imes c = c^{3}.

Step 5: Combine the Results

Combine the cubed coefficient and the cubed variable for the final expression: (-8) imes c^{3} = -8c^{3}. Therefore, (-2c)^{3} simplified is -8c^{3}.

Key Concepts

Simplifying expressions

When simplifying expressions, our main goal is to break down complex algebraic expressions into simpler and more manageable forms. This process often involves basic arithmetic operations and the application of mathematical rules for exponents.
Simplifying helps in understanding the expression better and makes further calculations easier.
In the provided exercise, simplifying the expression (-2c)^3 means identifying how each component of the parenthesis is affected by the exponent. Here, the expression consists of a number (-2) and a variable (c), both of which are in the same bracket.

• First, handle the numerical coefficient: cube it by multiplying it by itself three times.
• Then, manage the variable separately and apply the power to it as well.

Bringing it all together at the end gives the simplified version of the original expression.

Cubic powers

Cubic powers refer to raising an expression to the power of three. This means you are multiplying the expression by itself twice more. When you see something like (-2c)^3, it means you must apply the cube to both the number inside and the variable separately.
This operation results in each part being multiplied by itself three times. Cubic powers are significant because:

• They allow us to represent three-dimensional space in algebraic form. For example, calculating the volume of a cube involves cubing the length of one side.
• The rules applying to them are similar to those for other powers. This includes handling sign changes correctly based on whether the base is positive or negative.

For negative numbers like -2, observe that the cubic power maintains the negative sign. This is because an odd power of a negative number results in a negative product. Thus, in this exercise, -2 cubed equals -8, while c^3 is simply c multiplied by itself three times.

Algebraic expressions

Algebraic expressions include numbers, variables, and operations. They are the language used in algebra to describe different situations or problems. Understanding algebraic expressions involves:

• Recognizing components like constants (e.g., -2 in this exercise) and variables (e.g., c).
• Applying operations such as addition, subtraction, multiplication, division, and exponentiation.

In (-2c)^3, we tackle both the constant and the variable by applying the exponentiation operation to each.
This breaks down the problem into manageable parts, ensuring clarity while solving. Variables, like c, represent unknown values that can change, whereas -2 here is a fixed number or constant, defining a specific value in the expression. Different pieces of algebraic expressions like these may behave differently under mathematical operations, so handling each correctly is crucial. By effectively managing these elements, we simplify, cube, and ultimately understand their complete effect in algebraic contexts more clearly.