# Problem of the Day

If and represent positive real numbers and and , what is the value of ?

A. 2.6

B. 3.0

C. 3.4

D. 4.0

E. 4.2

### Solution

[latexpage]

First, square the expression $(a^4+b^2)^2$:

$(a^4+b^2)^2=$

$(a^4+b^2)(a^4+b^2)=$

$a^8+2b^2a^4+b^4$.

So now we have $a^8+2b^2a^4+b^4=198$

Rearranging the terms on the left:

$a^8+b^4+2b^2a^4=198$

Make a substitution with $a^8+b^4=36$ to get:

$36+2b^2a^4=198$

$2b^2a^4=162$

$b^2a^4=81$

Take the fourth root of both sides to get:

$a\sqrt{b}=3$

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