How do you rationalize the denominator and simplify #(2sqrt12 - sqrt5) /( sqrt5 + 4sqrt3)#?

Answer 1

See a solution process below:

To rationalize the fraction we need to use the rule:

#(color(red)(x) + color(blue)(y))(color(red)(x) - color(blue)(y)) = color(red)(x)^2 - color(blue)(y)^2#
Because the denominator is the form: #(color(red)(sqrt(5)) + color(blue)(4sqrt(3)))#
We need to use: #(color(red)(sqrt(5)) - color(blue)(4sqrt(3)))# as the numerator and denominator for the appropriate form of #1:
#(2sqrt(12) - sqrt(5))/((color(red)(sqrt(5)) + color(blue)(4sqrt(3)))) xx ((color(red)(sqrt(5)) - color(blue)(4sqrt(3))))/((color(red)(sqrt(5)) - color(blue)(4sqrt(3)))) =>#
#(2sqrt(12)sqrt(5) - (4 * 2)sqrt(12)sqrt(3) - (sqrt(5))^2 + 4sqrt(5)sqrt(3))/((color(red)(sqrt(5)))^2 - (color(blue)(4sqrt(3)))^2) =>#
#(2sqrt(60) - 8sqrt(36) - 5 + 4sqrt(15))/(5 - (16 * 3)) =>#
#(2sqrt(4 * 15) - (8 * 6) - 5 + 4sqrt(15))/(5 - 48) =>#
#(2sqrt(4)sqrt(15) - 48 - 5 + 4sqrt(15))/(-43) =>#
#((2 * 2)sqrt(15) - 53 + 4sqrt(15))/(-43) =>#
#(4sqrt(15) - 53 + 4sqrt(15))/(-43) =>#
#(-53 + 4sqrt(15) + 4sqrt(15))/(-43) =>#
#(-53 + 8sqrt(15))/(-43) =>#
#(-53)/(-43) + (8sqrt(15))/(-43) =>#
#53/43 - (8sqrt(15))/(43)#
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Answer 2

To rationalize the denominator and simplify the expression (2sqrt12 - sqrt5) /( sqrt5 + 4sqrt3), we can multiply both the numerator and denominator by the conjugate of the denominator, which is sqrt5 - 4sqrt3.

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

The simplified expression becomes:

[(2sqrt12 - sqrt5) * (sqrt5 - 4sqrt3)] / [(sqrt5 + 4sqrt3) * (sqrt5 - 4sqrt3)]

Simplifying further:

[(2sqrt12 * sqrt5) - (2sqrt12 * 4sqrt3) - (sqrt5 * sqrt5) + (sqrt5 * 4sqrt3)] / [(sqrt5 * sqrt5) - (4sqrt3 * sqrt5) + (4sqrt3 * sqrt5) - (4sqrt3 * 4sqrt3)]

This simplifies to:

[2sqrt60 - 8sqrt36 - 5 + 4sqrt15] / [5 - 48]

Further simplifying:

[2sqrt60 - 8sqrt36 - 5 + 4sqrt15] / [-43]

Finally, simplifying the expression:

(2sqrt60 - 8sqrt36 - 5 + 4sqrt15) / -43

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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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