Question

to expand the binomial expression \\((2x^3 + 3y^2)^7\\) using the binomial theorem, what should be substituted for the values of \\(a\\) and \\(b\\)?
\\(\bigcirc\\ a = x^{10}\\) and \\(b = y^9\\)
\\(\bigcirc\\ a = 2x^{10}\\) and \\(b = 3y^9\\)
\\(\bigcirc\\ a = x^3\\) and \\(b = y^2\\)
\\(\bigcirc\\ a = 2x^3\\) and \\(b = 3y^2\\)

Explanation:

Step1: Recall Binomial Theorem

The binomial theorem states that \((a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}\), where \(a\) and \(b\) are the two terms of the binomial.

Step2: Identify \(a\) and \(b\) in \((2x^{3}+3y^{2})^{7}\)

In the given binomial \((2x^{3}+3y^{2})^{7}\), comparing with \((a + b)^n\) (here \(n = 7\)), we can see that \(a=2x^{3}\) and \(b = 3y^{2}\). We check the options:

• Option 1: \(a=x^{10},b = y^{9}\) is incorrect.
• Option 2: \(a = 2x^{10},b=3y^{9}\) is incorrect.
• Option 3: \(a=x^{3},b = y^{2}\) misses the coefficients 2 and 3, so incorrect.
• Option 4: \(a = 2x^{3},b=3y^{2}\) matches our identification.

Answer:

D. \(a = 2x^{3}\) and \(b = 3y^{2}\) (assuming the last option is D, if the options are labeled as A - D with the last option being \(a = 2x^{3}\) and \(b = 3y^{2}\))