If 2^(4x) = 3,600, what is the value of \((2^{(1-\mathrm{x})})^2\) ?

1. Translate the problem requirements: We need to find the value of \((2^{1-x})^2\) given that \(2^{4x} = 3600\). This means we must first determine the value of x from the given equation, then use it to calculate the target expression.
2. Isolate the key relationship: Use the given equation \(2^{4x} = 3600\) to establish what \(2^x\) equals, since this will be the building block for our target expression.
3. Rewrite the target expression: Transform \((2^{1-x})^2\) into a form that uses \(2^x\), allowing us to substitute the value we found in the previous step.
4. Calculate the final result: Substitute the known value and perform the arithmetic to arrive at the answer.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're being asked to find. We have an equation involving powers of 2, and we need to use it to find the value of another expression.

Given information: \(2^{4x} = 3600\)
What we need to find: \((2^{1-x})^2\)

Think of this like having a puzzle piece (the given equation) that we need to use to complete another part of the puzzle (our target expression). The key is recognizing that both expressions involve powers of 2, so there must be a way to connect them.

Process Skill: TRANSLATE - Converting the problem statement into a clear mathematical relationship we can work with.

2. Isolate the key relationship

Now let's work with our given equation to find something useful. We have \(2^{4x} = 3600\).

Here's the key insight: if we can find what \(2^x\) equals, we'll be able to work with our target expression much more easily.

Notice that \(2^{4x}\) can be rewritten as \((2^x)^4\). This is because when we have a power raised to another power, we multiply the exponents: \(2^{4x} = 2^{x \times 4} = (2^x)^4\).

So our equation becomes: \((2^x)^4 = 3600\)

To find \(2^x\), we need to take the fourth root of both sides:
\(2^x = \sqrt[4]{3600}\)

Let's calculate this step by step. We need to find what number, when raised to the 4th power, gives us 3,600.

Let's factor 3,600: \(3600 = 36 \times 100 = 6^2 \times 10^2 = (6 \times 10)^2 = 60^2\)
So \(3600 = 60^2 = (60)^2\)

But we need the fourth root, so let's think about this differently:
\(3600 = 60^2 = (\sqrt{60})^4\)

To find \(\sqrt{60}\): \(60 = 4 \times 15\), so \(\sqrt{60} = 2\sqrt{15}\)

Therefore: \(2^x = 2\sqrt{15}\)

Process Skill: MANIPULATE - Using exponent rules to transform the equation into a more workable form.

3. Rewrite the target expression

Now let's work with our target expression: \((2^{1-x})^2\)

Using exponent rules, we can rewrite the exponent (1-x) as (1-x), and when we square the entire expression:
\((2^{1-x})^2 = 2^{2(1-x)} = 2^{2-2x}\)

We can further break this down:
\(2^{2-2x} = 2^2 \times 2^{-2x} = 4 \times 2^{-2x} = 4 \times (2^x)^{-2} = 4 \times \frac{1}{(2^x)^2}\)

So our target expression becomes: \(\frac{4}{(2^x)^2}\)

Process Skill: MANIPULATE - Transforming the target expression to use 2^x, which we found in the previous step.

4. Calculate the final result

Now we can substitute our known value. We found that \(2^x = 2\sqrt{15}\).

Our expression is: \(\frac{4}{(2^x)^2}\)

Substituting: \(\frac{4}{(2\sqrt{15})^2} = \frac{4}{4 \times 15} = \frac{4}{60} = \frac{1}{15}\)

Let's verify this makes sense by checking our work:

• We found \(2^x = 2\sqrt{15}\)
• \((2^x)^2 = (2\sqrt{15})^2 = 4 \times 15 = 60\)
• So \(\frac{4}{(2^x)^2} = \frac{4}{60} = \frac{1}{15}\)

Therefore, \((2^{1-x})^2 = \frac{1}{15}\)

5. Final Answer

The value of \((2^{1-x})^2 = \frac{1}{15}\)

This matches answer choice (B) 1/15.

To double-check: We used the given equation \(2^{4x} = 3600\) to find that \(2^x = 2\sqrt{15}\), then transformed our target expression to \(\frac{4}{(2^x)^2}\), which equals \(\frac{4}{60} = \frac{1}{15}\).

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the relationship between expressions: Students may try to directly substitute values or attempt algebraic manipulation without recognizing that both expressions involve powers of 2 with related exponents. They might miss that finding \(2^x\) is the key connecting link between the given equation and the target expression.

2. Incorrect exponent manipulation: Students may incorrectly rewrite \(2^{4x}\) as \(4 \times 2^x\) instead of \((2^x)^4\), leading them down an impossible solution path. This fundamental error in exponent rules makes the problem unsolvable using their approach.

Errors while executing the approach

1. Calculation errors when finding the fourth root: Students may incorrectly calculate \(\sqrt[4]{3600}\). Common errors include miscalculating the prime factorization of 3,600 or incorrectly simplifying \(\sqrt{60}\), leading to wrong values for \(2^x\).

2. Sign and exponent errors in algebraic manipulation: When transforming \((2^{1-x})^2\) to \(2^{2-2x}\) and then to \(\frac{4}{(2^x)^2}\), students frequently make errors with negative exponents or drop the coefficient 4, resulting in expressions like \(\frac{1}{(2^x)^2}\) instead of \(\frac{4}{(2^x)^2}\).

3. Arithmetic errors in final substitution: Even with correct intermediate steps, students may make computational errors when calculating \(\frac{4}{(2\sqrt{15})^2}\), such as incorrectly computing \((2\sqrt{15})^2 = 60\) or simplifying \(\frac{4}{60}\) to something other than \(\frac{1}{15}\).

Errors while selecting the answer

1. Selecting reciprocal or related values: Students who get intermediate results like 15, 60, or 1/(15) may mistakenly select answer choices that represent these intermediate calculations rather than the final answer 1/15. This often happens when they lose track of which expression they were originally asked to find.