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How do you use the binomial theorem to expand and simplify the expression #(1/x+y)^5#?

Answer 1

Serenity Johnson

#(1/x+y)^5=1/(x^5)+(5y)/x^4+(10y^2)/x^3+(10y^3)/x^2+(5y^4)/x+y^5#

#(1/x+y)^5#

=#(1/x)^5y^0*C(5,0)+(1/x)^4y^1*C(5,1)+(1/x)^3y^2*C(5,2)+(1/x)^2y^3*C(5,3)+(1/x)^1y^4*C(5,4)+(1/x)^0y^5*C(5,5)#

=#1/(x^5)+(5y)/x^4+(10y^2)/x^3+(10y^3)/x^2+(5y^4)/x+y^5#

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Answer 2

Benjamin Achtenberg

To expand and simplify the expression ( \left(\frac{1}{x} + y\right)^5 ) using the binomial theorem, you can use the formula:

[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k ]

Where:

( a = \frac{1}{x} )
( b = y )
( n = 5 )

Now, substitute these values into the formula and simplify.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.