Math Problem Statement

use the bionomial theorom to expand and simplify the following

Use the binomial theorem to expand and simplify the following expression: (2x + 3y)^6

Solution

The given expression is (2x+3y)6(2x + 3y)^6, and we are asked to expand it using the binomial theorem.

The binomial theorem states that:

(a+b)n=k=0n(nk)ankbk(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

In this case:

  • a=2xa = 2x
  • b=3yb = 3y
  • n=6n = 6

Thus, we apply the binomial expansion formula:

(2x+3y)6=k=06(6k)(2x)6k(3y)k(2x + 3y)^6 = \sum_{k=0}^{6} \binom{6}{k} (2x)^{6-k} (3y)^k

We will compute each term:

  1. For k=0k = 0:
    (60)(2x)6(3y)0=1(64x6)1=64x6\binom{6}{0} (2x)^6 (3y)^0 = 1 \cdot (64x^6) \cdot 1 = 64x^6

  2. For k=1k = 1:
    (61)(2x)5(3y)1=6(32x5)(3y)=576x5y\binom{6}{1} (2x)^5 (3y)^1 = 6 \cdot (32x^5) \cdot (3y) = 576x^5y

  3. For k=2k = 2:
    (62)(2x)4(3y)2=15(16x4)(9y2)=2160x4y2\binom{6}{2} (2x)^4 (3y)^2 = 15 \cdot (16x^4) \cdot (9y^2) = 2160x^4y^2

  4. For k=3k = 3:
    (63)(2x)3(3y)3=20(8x3)(27y3)=4320x3y3\binom{6}{3} (2x)^3 (3y)^3 = 20 \cdot (8x^3) \cdot (27y^3) = 4320x^3y^3

  5. For k=4k = 4:
    (64)(2x)2(3y)4=15(4x2)(81y4)=4860x2y4\binom{6}{4} (2x)^2 (3y)^4 = 15 \cdot (4x^2) \cdot (81y^4) = 4860x^2y^4

  6. For k=5k = 5:
    (65)(2x)1(3y)5=6(2x)(243y5)=2916xy5\binom{6}{5} (2x)^1 (3y)^5 = 6 \cdot (2x) \cdot (243y^5) = 2916xy^5

  7. For k=6k = 6:
    (66)(2x)0(3y)6=1(1)(729y6)=729y6\binom{6}{6} (2x)^0 (3y)^6 = 1 \cdot (1) \cdot (729y^6) = 729y^6

Now we combine all these terms to obtain the full expansion:

(2x+3y)6=64x6+576x5y+2160x4y2+4320x3y3+4860x2y4+2916xy5+729y6(2x + 3y)^6 = 64x^6 + 576x^5y + 2160x^4y^2 + 4320x^3y^3 + 4860x^2y^4 + 2916xy^5 + 729y^6

This is the expanded and simplified form of the expression.

Would you like more details on any part of this expansion? Let me know if you have questions!

Here are 5 related questions for practice:

  1. How would you expand (x+2y)5(x + 2y)^5 using the binomial theorem?
  2. What is the general formula for the binomial coefficient (nk)\binom{n}{k}?
  3. How would the expansion differ if the exponent in the original problem were 7 instead of 6?
  4. Can you apply the binomial theorem to expand (ab)6(a - b)^6?
  5. How can the binomial theorem be applied to solve real-world problems in probability or combinatorics?

Tip: The binomial coefficients (nk)\binom{n}{k} can also be found using Pascal's Triangle!

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Math Problem Analysis

Mathematical Concepts

Algebra
Binomial Expansion
Polynomials

Formulas

Binomial Expansion Formula: (a + b)^n = Σ(n choose k) * a^(n-k) * b^k

Theorems

Binomial Theorem

Suitable Grade Level

Grades 10-12