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Use special products to expand and simplify (2 + 2a)^6Show complete process and solution.

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spock2 | Student, Undergraduate | Honors

Posted November 1, 2011 at 2:36 PM via web

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Use special products to expand and simplify

(2 + 2a)^6

Show complete process and solution.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted November 30, 2011 at 3:02 AM (Answer #1)

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You could consider (2 + 2a)^6 = [(2 + 2a)^3]^2

Use the formula of binomial raised to cube:

(x+y)^3 = x^3 + 3x^2y + 3xy^2 + y^3

Use this formula to expand (2 + 2a)^3:

(2 + 2a)^3 = 2^3 + 3*4*2a + 3*2*4a^2 + 8a^3

(2 + 2a)^3 = 8 + 24a + 24a^2 + 8a^3

(2 + 2a)^3 = 8*(1+a^3) + 24a(1+a) = 8(1+a)(1 - a + a^2 + 3a)

(2 + 2a)^3 = 8(1+a)(a^2 + 2a + 1)

Use the special product (a+1)^2 instead of a^2 + 2a + 1.

(2 + 2a)^3 = 8(1+a)(1+a)^2 = 8(1+a)^3

Calculate (2 + 2a)^6 = [(2 + 2a)^3]^2 = [8(1+a)^3]^2 = 64(1+a)^6

Therefore, (2 + 2a)^6 = 64(1+a)^6

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