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WOMAN: Exercise 5.

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Expand

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A, (2x minus 3y)
all to the power of 4,

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B, (x minus 2 divided by x)
all to the power of 4.

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Part A.

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We use the binomial theorem,

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which states that (a + b)
all to the power of n

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is equal to the sum
of r equal to 0 up to r equal to n

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of n choose r
times a to the power of n minus r

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times b to the power of r.

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In this case, we wish
to let a = 2x,

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b = negative 3y

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and substituting these expressions
into the binomial theorem

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gives the following:

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And expanding the sum
gives the following:

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And then simplifying and simplifying
again gives the final expansion,

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which says that (2x – 3y)
all to the power of 4

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is equal to 16x to the power of 4

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minus 96x cubed y

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+ 216 x squared y squared

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minus 216 xy cubed

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and + 81y to the power of 4.

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Part B.

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Again we use the binomial theorem
and in this case we let a = x

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and b = negative 2 divided by x
and, again, n will equal 4.

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Substituting these expressions
into the binomial theorem

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and then expanding the sum

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and simplifying to give
the final expansion

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which says that x minus 2 divided
by x all to the power of 4

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is equal to x to the power of 4
minus 8x squared + 24

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minus 32 divided by x squared

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+ 16 divided by x to the power of 4.