How do you rationalize the denominator and simplify #(10-sqrt3)/(6+sqrt6)#?

Answer 1

#(60-10sqrt(6)-6sqrt(3)+3sqrt(2))/30#

#1#. Start by multiplying the numerator and denominator by the conjugate of the fraction's denominator, #6-sqrt(6)#.
#(10-sqrt(3))/(6+sqrt(6))#
#=(10-sqrt(3))/(6+sqrt(6))((6-sqrt(6))/(6-sqrt(6)))#
#2#. Simplify the numerator.
#=(60-10sqrt(6)-6sqrt(3)+sqrt(18))/(6+sqrt(6))(1/(6-sqrt(6)))#
#=(60-10sqrt(6)-6sqrt(3)+3sqrt(2))/(6+sqrt(6))(1/(6-sqrt(6)))#
#3#. Simplify the denominator. Note that it contains a difference of squares #(color(red)(a^2-b^2=(a+b)(a-b)))#.
#=(60-10sqrt(6)-6sqrt(3)+3sqrt(2))/(36-6)#
#=color(green)(|bar(ul(color(white)(a/a)(60-10sqrt(6)-6sqrt(3)+3sqrt(2))/30color(white)(a/a)|)))#
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Answer 2

To rationalize the denominator and simplify the expression (10 - √3) / (6 + √6), we can multiply both the numerator and denominator by the conjugate of the denominator, which is (6 - √6).

By doing this, we eliminate the square root in the denominator.

The simplified expression becomes:

[(10 - √3) * (6 - √6)] / [(6 + √6) * (6 - √6)]

Simplifying further:

[60 - 10√6 - 6√3 + √18] / [36 - 6√6 + 6√6 - √36]

Combining like terms:

[60 - 10√6 - 6√3 + √18] / [36 - √36]

Simplifying the square root:

[60 - 10√6 - 6√3 + 3√2] / [36 - 6]

Further simplification:

[54 - 10√6 - 6√3 + 3√2] / 30

This is the simplified expression after rationalizing the denominator.

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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.

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.

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.

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.

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