# Rationalize expression (5+2√6)/(7+4√3)

• MHB
• parry
In summary, rationalizing an expression means to eliminate any irrational numbers by manipulating the expression. This can be done by multiplying the expression by its conjugate, resulting in a difference of squares. Rationalizing expressions is important for simplifying and finding exact solutions. Not all expressions can be rationalized and there are alternative methods for rationalizing expressions.
parry
5+2 root 6 / 7+4root3 = a-b root3 find a and b

Hello and welcome to MHB! (Wave)

I've moved this thread to our elementary algebra forum, since this is a better fit for the problem and I've give the thread a descriptive title. We are given:

$$\displaystyle \frac{5+2\sqrt{6}}{7+4\sqrt{3}}=a-b\sqrt{3}$$

And instructed to find $a$ and $b$. Do you have any work to show or thoughts on how to begin?

Start by "rationalizing the denominator". That is, multiply both numerator and denominator of $$\displaystyle \frac{5+ 2\sqrt{6}}{7+ 4\sqrt{3}}$$ by $$\displaystyle 7- 4\sqrt{3}$$.

## 1. What does it mean to "rationalize" an expression?

Rationalizing an expression means to eliminate any irrational numbers in the expression, such as square roots or cube roots, by manipulating the expression in a way that the final result is a rational number (a number that can be expressed as a ratio of two integers).

## 2. How do I rationalize the expression (5+2√6)/(7+4√3)?

To rationalize this expression, we need to eliminate the irrational number in the denominator, 4√3. To do this, we can multiply both the numerator and denominator by the conjugate of 7+4√3, which is 7-4√3. This will result in a difference of squares, allowing us to eliminate the radical in the denominator. The final rationalized expression is (35-20√2)/(49-48) = (35-20√2)/1 = 35-20√2.

## 3. Why is it important to rationalize expressions?

Rationalizing expressions is important because it allows us to simplify and manipulate expressions in a way that is easier to work with and understand. It also allows us to find exact solutions to problems, rather than having to work with decimal approximations.

## 4. Can all expressions be rationalized?

No, not all expressions can be rationalized. Only expressions with irrational numbers in the denominator, such as square roots or cube roots, can be rationalized. Expressions with other types of irrational numbers, such as π or e, cannot be rationalized.

## 5. Are there any other methods to rationalize expressions?

Yes, there are other methods to rationalize expressions, such as using the rationalizing factor method or the completing the square method. These methods may be more useful in certain situations, so it is important to understand and be familiar with multiple methods of rationalizing expressions.

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