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If a and b represent positive real numbers and (a^4+b^2)^2=198 and a^8

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If a and b represent positive real numbers and (a^4+b^2)^2=198 and a^8  [#permalink]

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New post 02 Dec 2019, 23:37
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D
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Question Stats:

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Re: If a and b represent positive real numbers and (a^4+b^2)^2=198 and a^8  [#permalink]

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New post 03 Dec 2019, 01:38
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from above equations we get
a^4*b^2=81
a(sqrtb)= 3


OA:B
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Re: If a and b represent positive real numbers and (a^4+b^2)^2=198 and a^8  [#permalink]

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New post 03 Dec 2019, 02:02
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simplify

(a4+b2)2=198(a4+b2)2=198
a8+b^4+2a^4*b^2 = 198
given
a8+b4=36

2a^4*b^2 = 198-36
a^4*b^2= 81
square both sides
a^2*b=9
again square
a√b=3
IMO B

If a and b represent positive real numbers and (a4+b2)2=198(a4+b2)2=198 and a8+b4=36, what is the value of ab‾√ab?

A. 2.6
B. 3.0
C. 3.4
D. 4.0
E. 4.2
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Re: If a and b represent positive real numbers and (a^4+b^2)^2=198 and a^8  [#permalink]

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New post 03 Dec 2019, 04:36
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\((a^4+b^2)^2=198\)
\(a^8+b^4\)\(+2a^4.b^2=198\)
\(36\)\(+2a^4.b^2=198\)
\(2a^4.b^2=162\) --> \(a^4.b^2=81\)
\((a√b)^4=3^4\) --> \({a√b}=3\), because a and b are positive real numbers.

Final answer is (B)
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Re: If a and b represent positive real numbers and (a^4+b^2)^2=198 and a^8  [#permalink]

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New post 03 Dec 2019, 04:49
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Quote:
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


\(a^8+b^4=36\)
\((a^4+b^2)^2=198…a^8+b^4+2a^4b^2=198\)
\((36)+2a^4b^2=198…a^4b^2=81…a^2b=9…a\sqrt{b}=3\)

Ans (B)
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Re: If a and b represent positive real numbers and (a^4+b^2)^2=198 and a^8  [#permalink]

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New post 03 Dec 2019, 06:35
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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√b?

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

\((a^4 + b^2)^2 = 198\)
\(a^8+b^4 + 2.a^4.b^2 = 198\)
\(36 + 2.a^4.b^2 = 198\)
\(a^4.b^2 = 81\)
\(a^2.b = 9\)
\(a√b = - 3\) or \(3\)

Since a is positive and √b can't be negative, answer is 3.

Answer B.
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Re: If a and b represent positive real numbers and (a^4+b^2)^2=198 and a^8  [#permalink]

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New post 03 Dec 2019, 12:44
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Given that (a^4+b^2)^2=198 and a^8+b^4=36 we are to determine a√b.
(a^4+b^2)^2=a^8+2a^4b^2+b^4=198
but a^8+b^4=36
hence 36+2a^4b^2=198
2a^4b^2=162
a^4b^2=81
a^2b=9
a√b=3

The answer is therefore B.
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Re: If a and b represent positive real numbers and (a^4+b^2)^2=198 and a^8  [#permalink]

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New post 03 Dec 2019, 15:32
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a, b —real positive numbers

\((a^{4} +b^{2})^{2} = 198\)

\(a^{8} +b^{4} = 36\)

—> \(a*b^{1/2}\) = ???

\(a^{8} + 2a^{4}b^{2} + b^{4}= 198\)

\(a^{8} + b^{4}= 36\)

—> \(2a^{4}b^{2}= 162\)

\(a^{4}b^{2}= 81\)

—>\( (ab^{1/2})^{4}= 81\)

\(ab^{1/2}= 3\)

The answer is B

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Re: If a and b represent positive real numbers and (a^4+b^2)^2=198 and a^8  [#permalink]

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New post 05 Dec 2019, 20:23
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Bunuel wrote:

Competition Mode Question



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



Foiling the first equation, we have:

a^8 + b^4 + 2(a^4)(b^2) = 198

Since a^8 + b^4 = 36, we have:

36 + 2(a^4)(b^2) = 198

(a^4)(b^2) = 81

a^2 * b = 9

a√b = 3

Answer: B
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Re: If a and b represent positive real numbers and (a^4+b^2)^2=198 and a^8   [#permalink] 05 Dec 2019, 20:23
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