Question

which among the following ng has the least value
√75 - √74
√74-√73
√76 -√75
√72-
√ 73​

Answer 1

√76 - √75  has the least value in the given options.

Given:

• √75 - √74
•  √74 - √73
•  √76 - √75
•  √73 - √72​

To Find:

• Which option has Least Value

Solution:

Rationalizing factor :

The factor of multiplication by which an irrational number is multiplied to convert it into rational number

If the product of two irrational numbers or surds is a rational number, then each surd is a rationalizing factor for each other.

Step 1:

Rationalizing the numerator of  √75 - √74

$(\sqrt{75} - \sqrt{74} ) \times \dfrac{\sqrt{75} + \sqrt{74} }{\sqrt{75} + \sqrt{74} }$

Step 2:

Using identity (a + b)(a - b) = a² - b²    and  (√x)² =  x where x is non negative

$\dfrac{75 - 74 }{\sqrt{75} + \sqrt{74} }$

$\dfrac{1 }{\sqrt{75} + \sqrt{74} }$

Hence

$(\sqrt{75} - \sqrt{74} )= \dfrac{1 }{\sqrt{75} + \sqrt{74} }$

Step 3:

Similarly solving others

$(\sqrt{74} - \sqrt{73} )= \dfrac{1 }{\sqrt{74} + \sqrt{73} }$

$(\sqrt{76} - \sqrt{75} )= \dfrac{1 }{\sqrt{76} + \sqrt{75} }$

$(\sqrt{73} - \sqrt{72} )= \dfrac{1 }{\sqrt{73} + \sqrt{72} }$

Step 4:

As numerator is common  hence Least value have maximum denominator

Clearly √76  + √75 is the maximum

Hence √76  - √75  has the least value

Correct option is  √76 - √75 has the least value in the given options.