# Rationalizing denominators using conjugate surds: Why use fraction w/ num. = denom.?

#### Deano

##### New member
Hi, I am very new to this topic and I am having a difficult time trying to understand the concept behind this certain aspect of the topic Logarithms.

When trying to rationalize the denominator, I don't understand why I must multiply it by a fraction with the same numerator and denominator. Like the example here:

. . . . .$$\displaystyle \dfrac{3}{\sqrt{\strut 2\,}}\, =\, \dfrac{3}{\sqrt{\strut 2\,}}\, \left(\dfrac{\sqrt{\strut 2\,}}{\sqrt{\strut 2\,}}\right)\, =\, \dfrac{3}{2}\, \sqrt{\strut 2\,}$$

Surely i could just make the numerator 1 and the denominator square root of 2 right?

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#### Subhotosh Khan

##### Super Moderator
Staff member
Hi, I am very new to this topic and I am having a difficult time trying to understand the concept behind this certain aspect of the topic Logarithms.

When trying to rationalize the denominator, I don't understand why I must multiply it by a fraction with the same numerator and denominator. Like the example here:

. . . . .$$\displaystyle \dfrac{3}{\sqrt{\strut 2\,}}\, =\, \dfrac{3}{\sqrt{\strut 2\,}}\, \left(\dfrac{\sqrt{\strut 2\,}}{\sqrt{\strut 2\,}}\right)\, =\, \dfrac{3}{2}\, \sqrt{\strut 2\,}$$

Surely i could just make the numerator 1 and the denominator square root of 2 right?
When trying to rationalize the denominator, you don't want to change the value of your given expression. When you multiply it by a fraction with the same numerator and denominator, you are effectively multiplying by one (1). So the value of your original expression remain the same (in your case 3/√2) to justify the "=" sign.

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#### Jomo

##### Elite Member
Hi, I am very new to this topic and I am having a difficult time trying to understand the concept behind this certain aspect of the topic Logarithms.

When trying to rationalize the denominator, I don't understand why I must multiply it by a fraction with the same numerator and denominator. Like the example here:

. . . . .$$\displaystyle \dfrac{3}{\sqrt{\strut 2\,}}\, =\, \dfrac{3}{\sqrt{\strut 2\,}}\, \left(\dfrac{\sqrt{\strut 2\,}}{\sqrt{\strut 2\,}}\right)\, =\, \dfrac{3}{2}\, \sqrt{\strut 2\,}$$

Surely i could just make the numerator 1 and the denominator square root of 2 right?
Whenever you want to change the way somethings looks you multiply it by one in a favorable way. You need to understand this as it always works and get get you out of trouble. For example if you know that your car gets 20 miles per gallon (since 20 miles and 1 gallon are equal for your car, then 20miles/1gal = 1gal/20miles = 1), then you can easily compute how many gallons you need to go 220 miles.

220 miles = 220 miles * 1 = 220 miles * 1 gal/20 miles = 11 gallons. The conversion factor converts miles to gallons or gallons to miles.

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