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fourth roots

by westie » Fri Apr 10, 2015 11:26 pm
A magoosh DS question is "the square root of (x to the power of 4) =9. What is the value of x?
So I square both sides and get x to the power of 4 =81
so x=3,
because I thought that if the square root is written on the GMAT question, it implies positive roots only.
However, the answer is shown as not sufficient, because the roots are x= 3 and x=-3
So does the 'if the square root is written in the GMAT question, it implies the positive root only' only apply to square roots, and not other even roots?
many thanks for any help on this :)
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by GMATGuruNY » Sat Apr 11, 2015 2:41 am
What is the value of x?

Statement 1: √(x�) = 9
If we square both sides, we get:
[ √(x�) ]² = 9²
x� = 81.

The resulting equation does NOT have a root symbol.
Thus, when we solve for x, we must account for both the positive root AND the negative root:
(x�)^(1/4) = (81)^(1/4)
x = ±3.
Since it's possible that x=3 or that x=-3, the value of x cannot be determined.
INSUFFICIENT.
westie wrote:I thought that if the square root is written on the GMAT question, it implies positive roots only.
Correct:
√ = means the POSITIVE ROOT ONLY.
The equation in Statement 1 above abides by this rule:
√(x�) = 9.
Notice that √ on the left side yields ONLY A POSITIVE ROOT on the right side: 9.

This does not mean, however, that x itself must be positive.
As shown above, x=3 and x=-3 are both valid solutions for √(x�) = 9.
If we plug x=3 into √(x�) = 9, we get:
√(3�) = √81 = 9.
If we plug x=-3 into √(x�) = 9, we get:
√(-3)� = √81 = 9.
Thus, √(x�) = 9 implies that x = ±3.
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