MHB Simplifying an Expression: $\sqrt{3} + \sqrt{2} / \sqrt{3} - \sqrt{2}$

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The discussion focuses on simplifying the expression $\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}$ by rationalizing the denominator. To achieve this, one can multiply the expression by a form of 1, specifically $\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}+\sqrt{2}}$, which leads to a new expression with a rationalized denominator. The resulting form simplifies to $\frac{5 + 2\sqrt{6}}{1}$ after performing the necessary algebraic operations. Additionally, the thread suggests similar techniques for rationalizing both the numerator and denominator in general cases. The overall goal is to simplify the expression while eliminating the square roots from the denominator.
prasadini
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\sqrt{3} + \sqrt{2} / \sqrt{3} - \sqrt{2}
 
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Hello and welcome to MHB, prasadini! (Wave)

I've moved your thread to a more fitting area. :D

So, we are given the expressions (I assume):

$$\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}$$

I assume you are to rationalize the denominator...what form of $1$ do we need to multiply this expression by to accomplish this?
 
MarkFL said:
Hello and welcome to MHB, prasadini! (Wave)

I've moved your thread to a more fitting area. :D

So, we are given the expressions (I assume):

$$\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}$$

I assume you are to rationalize the denominator...what form of $1$ do we need to multiply this expression by to accomplish this?

$$\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}$$

is equal to



5+2√6 and 1 /5−2√6 How can i get this answer
 
prasadini said:
$$\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}$$

is equal to



5+2√6 and 1 /5−2√6 How can i get this answer

Well, suppose we are given:

$$\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}}$$

a) To rationalize the denominator, we would do the following:

$$\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}}\cdot\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}+\sqrt{b}}=\frac{(\sqrt{a}+\sqrt{b})^2}{a-b}=\frac{a+2\sqrt{ab}+b}{a-b}=\frac{a+b}{a-b}+\frac{2}{a-b}\sqrt{ab}$$

b) To rationalize the numerator, we would do the following:

$$\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}}\cdot\frac{\sqrt{a}-\sqrt{b}}{\sqrt{a}-\sqrt{b}}=\frac{a-b}{a-2\sqrt{ab}+b}=\frac{a-b}{a+b-2\sqrt{ab}}$$

Can you use these techniques to rationalize the denominator and numerator of the given expression?
 

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