Problem of the Day


If a and b represent positive real numbers and (a^4+b^2)^2=198 and a^8+b^4=36, what is the value of a\sqrt{b}?

A. 2.6
B. 3.0
C. 3.4
D. 4.0
E. 4.2

Reveal Answer

Answer

B. 3.0

See the Solution

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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