Given: \((5x)^{\frac{1}{8}} = (4x)^{\frac{1}{6}}\)

Let's dispense of these fractional exponents by raising both sides of the equation to the power of 24 (24 is the least common multiple of 8 and 6).

We get: \([(5x)^{\frac{1}{8}}]^{24} = [(4x)^{\frac{1}{6}}]^{24}\)

Apply the Power of a Power law to get: \((5x)^{3} = (4x)^{4}\)

Apply the Power of a Product law to get: \((5^3)(x^3) = (4^4)(x^4)\)

Evaluate: \(125x^3 = 256x^4\)

Divide both sides of the equation by \(x^3\) to get: \(125 = 256x\)

Divide both sides of the equation by \(256\) to get: \(\frac{125}{256} = x\)

So, \(x = \frac{125}{256}\) is ONE possible solution to the given equation.

At this point we need to recognize that \(x = 0\) is another possible solution since \(0^{\frac{1}{8}} = 0^{\frac{1}{6}}\)

So the range of all possible values of \(x = \frac{125}{256} - 0 = \frac{125}{256}\)

So we have:
QUANTITY A: \(\frac{125}{256}\)
QUANTITY B: \(0.5\)

No need to grab the calculator.
Since \(\frac{125}{250}=0.5\), we can conclude that \(\frac{125}{256}\) is less than \(0.5\)

Answer: B

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