Math Problem Statement

answer the questions

Find the product of (n + 13)^2 and (2k + 1)^3.

Solution

Let's solve the questions step by step.

Question 1:

You are asked to find the product of (n+13)2(n + 13)^2.

This expression is a perfect square trinomial, and it can be expanded using the formula: (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2 For (n+13)2(n + 13)^2:

  • a=na = n
  • b=13b = 13

Applying the formula: (n+13)2=n2+2(n)(13)+132(n + 13)^2 = n^2 + 2(n)(13) + 13^2 =n2+26n+169= n^2 + 26n + 169

So, the expanded form is: n2+26n+169n^2 + 26n + 169

Thus, the correct answer is the third option: n2+26n+169n^2 + 26n + 169.

Question 2:

You are asked to find the product of (2k+1)3(2k + 1)^3.

This is a cube expression and can be expanded using the formula: (a+b)3=a3+3a2b+3ab2+b3(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 For (2k+1)3(2k + 1)^3:

  • a=2ka = 2k
  • b=1b = 1

Applying the formula: (2k+1)3=(2k)3+3(2k)2(1)+3(2k)(12)+13(2k + 1)^3 = (2k)^3 + 3(2k)^2(1) + 3(2k)(1^2) + 1^3 =8k3+3(4k2)+3(2k)+1= 8k^3 + 3(4k^2) + 3(2k) + 1 =8k3+12k2+6k+1= 8k^3 + 12k^2 + 6k + 1

So, the expanded form is: 8k3+12k2+6k+18k^3 + 12k^2 + 6k + 1

If this option is available in the quiz, that is the correct answer.

Would you like further details on any step or have any questions about this? Here are five questions to deepen your understanding:

  1. How would the expansion change for (n13)2(n - 13)^2?
  2. What happens if we expand (2k1)3(2k - 1)^3 instead?
  3. Can you generalize the expansion of (a+b)2(a + b)^2 to higher powers?
  4. How does the binomial theorem apply to (2k+1)3(2k + 1)^3?
  5. What are some real-world applications of expanding binomials?

Tip: Always verify your answers by plugging in small values for nn or kk and checking if the results match.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomial Expansion
Binomial Theorem

Formulas

(a + b)^2 = a^2 + 2ab + b^2
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

Theorems

Binomial Theorem

Suitable Grade Level

Grade 9-11