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What is the numerical coefficient of the a^4b^4 term in the expansion of (13a^2 - 2b)^6?

Question:

What is the numerical coefficient of the {eq}a^4b^4 {/eq} term in the expansion of {eq}(13a^2 - 2b)^6? {/eq}

Binomial Expansion:

The binomial theorem is used in many application in mathematics. The binomial expansion of {eq}(x-y)^n = nC0 x^n y^0 -nC1 x^{n-1} y^1 +nC2 x^{n-2} y^2 -nC3 x^{n-3} y^3 + \cdots +(-1)^r nCr x^{n-r} y^r+\cdots {/eq}.

The general term is of the from {eq}T_{r+1}=(-1)^r nCr x^{n-r} y^r {/eq}.

To solve, we'll compute the general term of the given expansion by using the above formula.

Answer and Explanation:

We are given {eq}(13a^2 - 2b)^6 {/eq}


The general term is {eq}(x-y)^n {/eq} is {eq}T_{r+1}=(-1)^r nCr x^{n-r} y^r. {/eq}


In this problem, {eq}x=13a^2 . y = 2b , n=6 , r = 4 {/eq}

{eq}T_{4+1}=(-1)^4 6C4 (13a^2)^{6-4} (2b)^4 {/eq}

{eq}\Rightarrow T_{5}=15 (13a^2)^{2} (2b)^4 {/eq}

{eq}\Rightarrow T_{5}=15 \cdot 13^2 \cdot 2^4 a^4 b^4 {/eq}

{eq}\Rightarrow T_{5}=40560 a^4 b^4 {/eq}


Therefore, the numerical coefficient of the a^4b^4 term in the expansion is {eq}= 40560 {/eq}.


Learn more about this topic:

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The Binomial Theorem: Defining Expressions

from Algebra II: High School

Chapter 12 / Lesson 7
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