Question

For the given term, find the binomial raised to the power, whose expansion it came from: 220(5x)3(−6y)9.

Answer 1

Answer: $(5x - 6y)^{12}$
Explanation: For a general binomial expansion, $(x + y)^{n}$, we know that the powers have to add up to the initial power. This means that the power of x and power of y have to add up to n. This is the binomial theorem.

To further demonstrate this, let's use:
$(x + y)^{4}$

We can easily expand this. Using Pascal's Triangle, we get:
$(x + y)^{4} = x^{4} \cdot y^{0} + 4x^{3} \cdot y^{1} + 6x^{2} \cdot y^{2} + 4x^{1} \cdot y^{3} + x^{0} \cdot y^{4}$

As we progress along the expansion, we can see that in each term, the summation of each power remains constant, namely 4.

It doesn't matter what term the binomials are, because the power summation will never change.
This is why we can say that it is raised to the 12th power, and the binomial is:
(5x - 6y).

Thus, we get: $(5x - 6y)^{12}$