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How do you simplify #5/(4 + sqrt5)#?

Answer 1

See a solution process below:

To simplify this express we need to rationalize the denominator. Or, in other words, remove the radicals from the denominator. We can do this by multiplying by the appropriate form of #1#:

#(4 - sqrt(5))/(4 - sqrt(5)) xx 5/(4 + sqrt(5)) => (5(4 - sqrt(5)))/(4^2 + 4sqrt(5) - 4sqrt(5) + (sqrt(5))^2) =>#

#((5 xx 4) - (5 xx sqrt(5)))/(16 + (4sqrt(5) - 4sqrt(5)) + 5) =>#

#(20 - 5sqrt(5))/(16 + 0 + 5) =>#

#(20 - 5sqrt(5))/21#

#20/21 - (5sqrt(5))/21#

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Answer 2

Hazel Anglin

To simplify 5/(4 + sqrt5), we need to rationalize the denominator. Multiply both the numerator and denominator by the conjugate of the denominator, which is 4 - sqrt5. Simplifying this expression gives us (5 * (4 - sqrt5)) / ((4 + sqrt5) * (4 - sqrt5)). Expanding and simplifying further, we get (20 - 5sqrt5) / (16 - 5). Finally, simplifying the denominator gives us (20 - 5sqrt5) / 11.

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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.