Math Problem Statement
Suppose n is a natural number and n>2. Prove the result of n to the power of 3 minus 2n power of 2
Solution
We are tasked with proving the expression:
where is a natural number and .
Step-by-Step Breakdown:
Let's first factor the expression to simplify it.
This factorization can be interpreted as follows:
- is a common factor between both terms.
- The remaining factor is , resulting in the final form .
Analysis of the Expression:
Given that is a natural number and , we can now evaluate the nature of this expression.
-
For :
-
For :
-
For :
This shows that the expression produces positive results for all , and the result grows as increases.
General Conclusion:
The expression is valid for all , , and yields positive values.
Would you like further clarification on any part of this explanation?
Here are 5 related questions for practice:
- What happens when in the expression ?
- How does the behavior of the expression change for ?
- Factor the expression .
- Solve the inequality for .
- How would the expression behave for any constant ?
Tip: Always look for common factors first when simplifying algebraic expressions—it can make problem-solving more efficient!
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Math Problem Analysis
Mathematical Concepts
Algebra
Exponents
Factoring
Formulas
n^3 - 2n^2 = n^2(n - 2)
Theorems
Factoring
Suitable Grade Level
Grades 8-10