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:

n32n2n^3 - 2n^2 where nn is a natural number and n>2n > 2.

Step-by-Step Breakdown:

Let's first factor the expression to simplify it.

n32n2=n2(n2)n^3 - 2n^2 = n^2(n - 2)

This factorization can be interpreted as follows:

  • n2n^2 is a common factor between both terms.
  • The remaining factor is n2n - 2, resulting in the final form n2(n2)n^2(n - 2).

Analysis of the Expression:

Given that nn is a natural number and n>2n > 2, we can now evaluate the nature of this expression.

  1. For n=3n = 3: n32n2=332(32)=2718=9n^3 - 2n^2 = 3^3 - 2(3^2) = 27 - 18 = 9

  2. For n=4n = 4: n32n2=432(42)=6432=32n^3 - 2n^2 = 4^3 - 2(4^2) = 64 - 32 = 32

  3. For n=5n = 5: n32n2=532(52)=12550=75n^3 - 2n^2 = 5^3 - 2(5^2) = 125 - 50 = 75

This shows that the expression produces positive results for all n>2n > 2, and the result grows as nn increases.

General Conclusion:

The expression n32n2=n2(n2)n^3 - 2n^2 = n^2(n - 2) is valid for all nNn \in \mathbb{N}, n>2n > 2, and yields positive values.

Would you like further clarification on any part of this explanation?


Here are 5 related questions for practice:

  1. What happens when n=2n = 2 in the expression n32n2n^3 - 2n^2?
  2. How does the behavior of the expression change for n2n \leq 2?
  3. Factor the expression n43n3+2n2n^4 - 3n^3 + 2n^2.
  4. Solve the inequality n32n2>0n^3 - 2n^2 > 0 for nNn \in \mathbb{N}.
  5. How would the expression n3kn2n^3 - kn^2 behave for any constant kk?

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