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Use Pascal's Triangle to expand
(4y^(2)+5x^(2))^(4). Express your answer in simplest form.
Answer:

Use Pascal's Triangle to expand $\left(4 y^{2}+5 x^{2}\right)^{4}$ . Express your answer in simplest form. $\newline$ Answer:

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Pascal's triangle and the Binomial Theorem

Full solution

Q. Use Pascal's Triangle to expand

\left(4 y^{2}+5 x^{2}\right)^{4}

. Express your answer in simplest form.

\newline

Answer:

Identify Row: Identify the row of Pascal's Triangle that corresponds to the exponent $4$ . The row for the exponent $4$ is the fifth row (starting with row $0$ for the exponent $0$ ), which has the numbers $1, 4, 6, 4, 1$ .
Write Expansion Terms: Write out the terms of the expansion using the coefficients from Pascal's Triangle. $\newline$ The expansion will have the form: $\newline$ $1\times(4y^2)^4\times(5x^2)^0 + 4\times(4y^2)^3\times(5x^2)^1 + 6\times(4y^2)^2\times(5x^2)^2 + 4\times(4y^2)^1\times(5x^2)^3 + 1\times(4y^2)^0\times(5x^2)^4$ .
Calculate Each Term: Calculate each term of the expansion. $\newline$ $1$ st term: $1*(4y^2)^4*(5x^2)^0 = 1*256y^8*1 = 256y^8$ $\newline$ $2$ nd term: $4*(4y^2)^3*(5x^2)^1 = 4*64y^6*5x^2 = 1280y^6x^2$ $\newline$ $3$ rd term: $6*(4y^2)^2*(5x^2)^2 = 6*16y^4*25x^4 = 2400y^4x^4$ $\newline$ $4$ th term: $4*(4y^2)^1*(5x^2)^3 = 4*4y^2*125x^6 = 2000y^2x^6$ $\newline$ $5$ th term: $1*(4y^2)^0*(5x^2)^4 = 1*1*625x^8 = 625x^8$
Combine Terms: Combine all the terms to write the final expanded form. $\newline$ The expanded form is: $\newline$ $256y^8 + 1280y^6x^2 + 2400y^4x^4 + 2000y^2x^6 + 625x^8$ .

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