Question

Translate each sentence into an equation. (Lesson 2-1). (p. 140)

One half of m cubed is the same as four times m minus nine.

Short answer

The required equation is $\frac{1}{2}{m}^3=4m-9$.

Step-by-step solution

Step 1. Apply the concept of raising variable to an index.

When a variable is multiplied by itself we raise the index/exponent of the variable by two, when it is multiplied once more to this product we raise the index to three, and so on.

In this fashion variables are written, for example:

$\begin{array}{l}x\times x=x^2\\ x\times x=x^3\end{array}$

Thus$m$ cubed is$m$ raised to the third power.

$m^3$and its half is:

$\frac{1}{2}{m}^3$

Step 2. Apply the concept of multiplication of a constant and a variable.

When a constant is multiplied by a variable we just append the constant in front of the variable.

We are given a constant $4$and a variable $m$, so their product will be$4m$

Step 3. Apply the concept of difference for unlike terms.

Like terms are the terms having the same variables and being raised to the same index. A coefficient is the appended number in front of the variable; if no number is shown then we assume it to be 1.

For example,$4x$ and$5x$ are like terms.

Unlike terms are the terms not having the same variables and or are raised to a different index.

For example, 4 and$5x$ are unlike terms.

We can subtract like terms by subtracting their coefficients, we cannot subtract the unlike terms.

Now we are given an expression$4m$ and we subtract 9 from it to obtain

$4m-9$

Step 4. Evaluate the two results.

To get the final expression we equate the results of steps 1 and 2 to get:

$\frac{1}{2}{m}^3=4m-9$

Thus, the required equation is $\frac{1}{2}{m}^3=4m-9$.