Question

Find the smallest number that a√+b√−−−−−−−√−b√−a√−−−−−−−√ must be multiplied by so that the result is a perfect square.

Answer 1

The smallest number is $\frac{a+\sqrt{b} }{a+\sqrt{b} }$ which is 1.

• A perfect square is a number that can be written as the second exponent of an integer or as the product of an integer by itself. When you multiply an integer by itself, you get a perfect square, which is a positive integer. Perfect squares are sums that are the products of integers multiplied by themselves, to put it simply.
• By squaring a whole number or an integer, one can create perfect squares.

Here, according to the given information, it is given that,

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

Now, this will become a perfect square, if we rationalize the denominator.

Now, we get,

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

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

Hence, the smallest number is $\frac{a+\sqrt{b} }{a+\sqrt{b} }$ which is 1.

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