Let's break down the given mathematical expression step-by-step to interpret it correctly.
The given expression is:
[tex]\[
\frac{(5x-2)^3}{x+11}
\][/tex]
We need to describe this expression in a structured way using specific terms for each part.
1. First Part: [tex]\((5x - 2)^3\)[/tex]
- Here, the term inside the parentheses, [tex]\(5x - 2\)[/tex], is considered as one unit.
- The expression inside the parentheses is raised to the power of 3, which is referred to as the "cube".
- The terms [tex]\(5x\)[/tex] and 2 are involved in a subtraction, so it is described as a "difference".
2. Second Part: [tex]\(x + 11\)[/tex]
- The terms [tex]\(x\)[/tex] and 11 are combined by addition, so it is referred to as a "sum".
Based on this breakdown, we need to fill in the blanks:
[tex]\[
\text{the }\square\text{ of the }\square\text{ of the }\square\text{ of }5x\text{ and }2\text{ and the }\square\text{ of }x\text{ and }11
\][/tex]
Let's fill in the blanks correctly:
1. The first blank should refer to "cube" because we are dealing with the third power of something.
2. The second blank should be "difference" because [tex]\(5x - 2\)[/tex] involves subtraction.
3. The third blank should cover the terms involved in the difference, which are "5x and 2".
4. The fourth blank should be "sum" because [tex]\(x + 11\)[/tex] involves addition.
5. The fifth blank should be the terms involved in the sum, which are "x and 11".
So, the blanks are filled as:
[tex]\[
\text{the \textbf{cube} of the \textbf{difference} of the \textbf{5x and 2} and the \textbf{sum} of \textbf{x and 11}}
\][/tex]
Thus, the correct interpretation of the given expression is:
1. the cube
2. of the difference
3. of the 5x and 2
4. and the sum
5. of x and 11.
