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use the bionomial theorom to expand and simplify the following 

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

The binomial theorem states that:

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

In this case:
- \(a = 2x\)
- \(b = 3y\)
- \(n = 6\)

Thus, we apply the binomial expansion formula:

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

We will compute each term:

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

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

3. For \(k = 2\):  
   \[
   \binom{6}{2} (2x)^4 (3y)^2 = 15 \cdot (16x^4) \cdot (9y^2) = 2160x^4y^2
   \]

4. For \(k = 3\):  
   \[
   \binom{6}{3} (2x)^3 (3y)^3 = 20 \cdot (8x^3) \cdot (27y^3) = 4320x^3y^3
   \]

5. For \(k = 4\):  
   \[
   \binom{6}{4} (2x)^2 (3y)^4 = 15 \cdot (4x^2) \cdot (81y^4) = 4860x^2y^4
   \]

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

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

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

\[
(2x + 3y)^6 = 64x^6 + 576x^5y + 2160x^4y^2 + 4320x^3y^3 + 4860x^2y^4 + 2916xy^5 + 729y^6
\]

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 \((x + 2y)^5\) using the binomial theorem?
2. What is the general formula for the binomial coefficient \(\binom{n}{k}\)?
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 \((a - b)^6\)?
5. How can the binomial theorem be applied to solve real-world problems in probability or combinatorics?

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

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

Learn how to expand and simplify the expression (2x + 3y)^6 using the binomial theorem. This step-by-step solution covers binomial expansion and polynomial terms, suitable for students in Grades 10-12.

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