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Question

Use the binomial theorem to expand the expressions: left parenthesis 2 m plus 2 n right parenthesis to the power of 6

hintHint:

The binomial expansion is left parenthesis x plus y right parenthesis to the power of n equals sum from k equals 0 to n of   n C subscript k x to the power of n minus k end exponent y to the power of k comma text  here  end text n greater or equal than 0 .
We are asked to find the expansion of left parenthesis 2 m plus 2 n right parenthesis to the power of 6 using Binomial theorem.

The correct answer is: (n+1)th row of the triangle.


     Step 1 of 2:
    The given expression is left parenthesis 2 m plus 2 n right parenthesis to the power of 6 . Here, the value of n=6, so we would have 6+1=7 terms in the expansion of the expression. The values of x equals 2 m straight & y equals 2 n .
    Step 2 of 2:
    Substitute the values in the binomial equation to get the terms of the expansion:

    table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell left parenthesis 2 m plus 2 n right parenthesis to the power of 6 equals 6 C subscript 0 left parenthesis 2 m right parenthesis to the power of 6 plus 6 C subscript 1 left parenthesis 2 m right parenthesis to the power of 5 left parenthesis 2 n right parenthesis plus 6 C subscript 2 left parenthesis 2 m right parenthesis to the power of 4 left parenthesis 2 n right parenthesis squared plus 6 C subscript 3 left parenthesis 2 m right parenthesis cubed left parenthesis 2 n right parenthesis cubed plus end cell row cell 6 C subscript 4 left parenthesis 2 m right parenthesis squared left parenthesis 2 n right parenthesis to the power of 4 plus 6 C subscript 5 left parenthesis 2 m right parenthesis left parenthesis 2 n right parenthesis to the power of 5 plus 6 C subscript 6 left parenthesis 2 n right parenthesis to the power of 6 end cell row cell equals 64 m to the power of 6 plus 6 open parentheses 32 m to the power of 5 close parentheses left parenthesis 2 n right parenthesis plus 15 open parentheses 16 m to the power of 4 close parentheses open parentheses 4 n squared close parentheses plus 20 open parentheses 8 m cubed close parentheses open parentheses 8 n cubed close parentheses plus end cell row cell 15 open parentheses 4 m squared close parentheses open parentheses 16 n to the power of 4 close parentheses plus 6 left parenthesis 2 m right parenthesis open parentheses 32 n to the power of 5 close parentheses plus 64 n to the power of 6 end cell row cell equals 64 m to the power of 6 plus 384 m to the power of 5 n plus 960 m to the power of 4 n squared plus 1280 m cubed n cubed plus 960 m squared n to the power of 4 plus 384 m n to the power of 5 plus 64 n to the power of 6 end cell end table
    Thus, the expansion of the expression is; left parenthesis 2 m plus 2 n right parenthesis to the power of 6 equals 64 m to the power of 6 plus 384 m to the power of 5 n plus 960 m to the power of 4 n squared plus 1280 m cubed n cubed plus 960 m squared n to the power of 4 plus 384 m n to the power of 5 plus 64 n to the power of 6

    The answer can also be found using the Pascal’s triangle. For the expansion of the expression (x + y)n , we would consider the (n+1)th row in the triangle.

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