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How do you rationalize the denominator and simplify #1/ (x(1-sqrt x) ) #?

Answer 1

#1/(x(1-sqrtx))=(1+sqrtx)/(x(1-x))#

To rationalize the denominator and simplify #1/(x(1-sqrtx))#, one needs to multiply the numerator and denominator, by the conjugate of #(1-sqrtx)#, i.e. by #(1+sqrtx)#.

Hence,

#1/(x(1-sqrtx))=(1+sqrtx)/(x(1-sqrtx)(1+sqrtx))#

= #(1+sqrtx)/(x(1-x))#

Note here we have avoided #x#, as it is a rational expression.

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Answer 2

Hazel Anglin

To rationalize the denominator and simplify the expression 1/(x(1-sqrt(x))), we can multiply both the numerator and denominator by the conjugate of the denominator, which is (1+sqrt(x)). This will eliminate the square root in the denominator.

By multiplying the numerator and denominator by (1+sqrt(x)), we get:

1/(x(1-sqrt(x))) * (1+sqrt(x))/(1+sqrt(x))

Simplifying this expression, we have:

(1+sqrt(x))/(x(1-sqrt(x))(1+sqrt(x)))

Expanding the denominator, we get:

(1+sqrt(x))/(x(1-sqrt(x)+sqrt(x)-x))

Simplifying further, we have:

(1+sqrt(x))/(x(1-x))

Therefore, the rationalized and simplified expression is (1+sqrt(x))/(x(1-x)).

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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.