If x > 0 and y > 0 and \((\mathrm{x}^{\frac{1}{6}})(\mathrm{y}^2)=\mathrm{y}^{\frac{4}{3}}\), which of the following is equivalent to x?

1. Translate the problem requirements: We have an equation with fractional exponents where x and y are positive, and we need to express x in terms of y by isolating x on one side of the equation.
2. Isolate the x term: Move all y terms to the right side of the equation to get \(\mathrm{x}^{1/6}\) by itself on the left side.
3. Eliminate the fractional exponent on x: Raise both sides to the 6th power to convert \(\mathrm{x}^{1/6}\) into x, since \((\mathrm{x}^{1/6})^6 = \mathrm{x}\).
4. Simplify the right side using exponent rules: Apply the power rule \((\mathrm{a}^m)^n = \mathrm{a}^{mn}\) to combine and simplify the y terms into a single expression.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we have and what we need to find. We're given an equation with two positive numbers x and y that have fractional exponents: \((\mathrm{x}^{1/6})(\mathrm{y}^2) = \mathrm{y}^{4/3}\). Our job is to rearrange this equation so that x is all by itself on one side, expressed only in terms of y.

Think of this like solving any equation - we want to get x alone. The tricky part is that x has a fractional exponent of 1/6, which means we're dealing with the sixth root of x.

Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding

2. Isolate the x term

Right now our equation is \((\mathrm{x}^{1/6})(\mathrm{y}^2) = \mathrm{y}^{4/3}\). We want to get \(\mathrm{x}^{1/6}\) by itself on the left side.

To do this, we need to move that \(\mathrm{y}^2\) term away from \(\mathrm{x}^{1/6}\). Since they're being multiplied together, we can divide both sides by \(\mathrm{y}^2\):

\((\mathrm{x}^{1/6})(\mathrm{y}^2) ÷ \mathrm{y}^2 = \mathrm{y}^{4/3} ÷ \mathrm{y}^2\)

On the left side, the \(\mathrm{y}^2\) terms cancel out, leaving us with just \(\mathrm{x}^{1/6}\).
On the right side, we use the rule that when we divide powers with the same base, we subtract the exponents: \(\mathrm{y}^{4/3} ÷ \mathrm{y}^2 = \mathrm{y}^{4/3 - 2} = \mathrm{y}^{4/3 - 6/3} = \mathrm{y}^{-2/3}\)

So now we have: \(\mathrm{x}^{1/6} = \mathrm{y}^{-2/3}\)

3. Eliminate the fractional exponent on x

Now we need to get rid of that fractional exponent on x. Since \(\mathrm{x}^{1/6}\) means "the sixth root of x," to get just x, we need to raise both sides to the 6th power.

When we raise both sides to the 6th power:
\((\mathrm{x}^{1/6})^6 = (\mathrm{y}^{-2/3})^6\)

On the left side, we use the power rule: when we raise a power to another power, we multiply the exponents. So \((\mathrm{x}^{1/6})^6 = \mathrm{x}^{(1/6) × 6} = \mathrm{x}^1 = \mathrm{x}\)

On the right side, we apply the same rule: \((\mathrm{y}^{-2/3})^6 = \mathrm{y}^{(-2/3) × 6} = \mathrm{y}^{-4}\)

Process Skill: MANIPULATE - Using exponent rules systematically to isolate the variable

4. Simplify the right side using exponent rules

We've already applied the key exponent rule in the previous step, but let's verify our work:

Starting from \(\mathrm{x}^{1/6} = \mathrm{y}^{-2/3}\), when we raised both sides to the 6th power:

• Left side: \((\mathrm{x}^{1/6})^6 = \mathrm{x}^{1/6 × 6} = \mathrm{x}^1 = \mathrm{x}\)
• Right side: \((\mathrm{y}^{-2/3})^6 = \mathrm{y}^{-2/3 × 6} = \mathrm{y}^{-4}\)

Therefore: \(\mathrm{x} = \mathrm{y}^{-4}\)

Final Answer

Our final result is \(\mathrm{x} = \mathrm{y}^{-4}\), which matches answer choice A: \(\mathrm{y}^{-4}\).

Let's verify this makes sense by checking our work backwards: if \(\mathrm{x} = \mathrm{y}^{-4}\), then \(\mathrm{x}^{1/6} = (\mathrm{y}^{-4})^{1/6} = \mathrm{y}^{-4/6} = \mathrm{y}^{-2/3}\). Substituting into our original equation: \((\mathrm{y}^{-2/3})(\mathrm{y}^2) = \mathrm{y}^{-2/3 + 2} = \mathrm{y}^{4/3}\) ✓

The answer is A.

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misunderstanding the goal of the problem
Students may think they need to find the value of x numerically rather than expressing x in terms of y. This leads them to try substituting specific values or looking for a single number answer, when the problem actually requires algebraic manipulation to express one variable in terms of another.

Faltering Point 2: Intimidation by fractional exponents
Many students see fractional exponents like \(\mathrm{x}^{1/6}\) and \(\mathrm{y}^{4/3}\) and either panic or try to convert everything to radicals unnecessarily. This can lead them to avoid the systematic algebraic approach needed, or make the problem more complicated than it needs to be.

Errors while executing the approach

Faltering Point 1: Incorrect application of exponent rules when dividing
When dividing \(\mathrm{y}^{4/3}\) by \(\mathrm{y}^2\), students often make errors with the subtraction of exponents. They might incorrectly calculate \(4/3 - 2\) as \(4/3 - 2/1 = 2/3\) instead of converting to common denominators: \(4/3 - 6/3 = -2/3\). This sign error is particularly common.

Faltering Point 2: Mistakes when raising both sides to the 6th power
When eliminating the fractional exponent by raising both sides to the 6th power, students frequently make errors in multiplying exponents. They might incorrectly calculate \((-2/3) × 6\) as \(-12/3 = -4\), getting the wrong final answer, or forget to apply the power to both sides of the equation.

Faltering Point 3: Arithmetic errors with fractional exponent calculations
Students often struggle with the mental math involved in fractional exponents, such as calculating \((1/6) × 6 = 1\) or \((-2/3) × 6 = -4\). These seemingly simple calculations become error-prone when working with fractions under time pressure.

Errors while selecting the answer

Faltering Point 1: Confusion between \(\mathrm{y}^{-4}\) and \(\mathrm{y}^4\)
Even after correctly calculating that \(\mathrm{x} = \mathrm{y}^{-4}\), students might select answer choice D (\(\mathrm{y}^4\)) instead of A (\(\mathrm{y}^{-4}\)) because they overlook or misinterpret the negative exponent. The negative exponent notation can be easily missed when scanning answer choices quickly.