Answer :

Answer:

[tex]2+\frac{5}{6} \sqrt{6}[/tex], where a = 2, b = [tex]-\frac{5}{6}[/tex] and c = 0.

Step-by-step explanation:

Recall that rationalizing a fraction means to use the radical conjugate of the denominator to "eliminate" the radicals in the denominator.  This "elimination" is needed due to the fact that radicals are irrational numbers thus making them not ideal for a denominator value.

Before identifying and multiplying the numerator and denominator by the conjugate, [tex]\sqrt{18} -\sqrt{12}[/tex] can be simplified to [tex]3\sqrt{2} -2\sqrt{3}[/tex] using the tree method.

The radical conjugate of the denominator value is [tex]\rm \sqrt{18} +\sqrt{12} \:\:or\:\:\ 3\sqrt{2} +2\sqrt{3}[/tex].

Multiplying the denominator by the conjugate the new denominator value is,

[tex](\sqrt{18} -\sqrt{12} )(\sqrt{18} +\sqrt{12} )=18-12[/tex].

This multiplication expression takes on the form of difference of squares formula of [tex](a-b)(a+b)=a^2-b^2[/tex] and since any radical squared is just the term underneath the radical, the expression is easily computed to 18 - 12.

Multiplying the numerator by the conjugate the new numerator value is,

[tex](\sqrt{2} +\sqrt{3} )(3\sqrt{2} +2\sqrt{3} )=3(\sqrt{2} ^2)+3\sqrt{2*3} +2\sqrt{3*2} +2(\sqrt{3} ^2)\\\\=6+3\sqrt{6} +2\sqrt{6} +6\\\\\implies12+5\sqrt{6}[/tex].

So, the fraction reduces to,

[tex]\frac{12+5\sqrt{6} }{18-12} =\frac{12+5\sqrt{6}}{6} =\frac{12}{6} +\frac{5\sqrt{6} }{6} \\\\\implies2+\frac{5}{6} \sqrt{6}[/tex].

2 correlates to the a term that doesn't have a radical multiplied to it, 0 correlates to the c term since the final answer doesn't include a radical 3 anywhere.

- [tex]\frac{5}{6}[/tex] correlates to the b term, since there's a negative sign next to the b term and our final answer's radical 6 term is positive, the [tex]\frac{5}{6}[/tex] term must be negative as a negative multiplied by a negative equals a positive.

Click here to learn more about radical conjugates:

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