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, and we are asked to expand it using the binomial theorem.
The binomial theorem states that:
(a+b)n=∑k=0n(kn)an−kbk
In this case:
- a=2x
- b=3y
- n=6
Thus, we apply the binomial expansion formula:
(2x+3y)6=∑k=06(k6)(2x)6−k(3y)k
We will compute each term:
-
For k=0:
(06)(2x)6(3y)0=1⋅(64x6)⋅1=64x6
-
For k=1:
(16)(2x)5(3y)1=6⋅(32x5)⋅(3y)=576x5y
-
For k=2:
(26)(2x)4(3y)2=15⋅(16x4)⋅(9y2)=2160x4y2
-
For k=3:
(36)(2x)3(3y)3=20⋅(8x3)⋅(27y3)=4320x3y3
-
For k=4:
(46)(2x)2(3y)4=15⋅(4x2)⋅(81y4)=4860x2y4
-
For k=5:
(56)(2x)1(3y)5=6⋅(2x)⋅(243y5)=2916xy5
-
For k=6:
(66)(2x)0(3y)6=1⋅(1)⋅(729y6)=729y6
Now we combine all these terms to obtain the full expansion:
(2x+3y)6=64x6+576x5y+2160x4y2+4320x3y3+4860x2y4+2916xy5+729y6
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:
- How would you expand (x+2y)5 using the binomial theorem?
- What is the general formula for the binomial coefficient (kn)?
- How would the expansion differ if the exponent in the original problem were 7 instead of 6?
- Can you apply the binomial theorem to expand (a−b)6?
- How can the binomial theorem be applied to solve real-world problems in probability or combinatorics?
Tip: The binomial coefficients (kn) can also be found using Pascal's Triangle!