If t = 2^(x) + 1, then in terms of t, 4^(x) is

1. Translate the problem requirements: We're given that \(\mathrm{t = 2^{(x+1)}}\) and need to express \(\mathrm{4^x}\) in terms of t (meaning we need to rewrite \(\mathrm{4^x}\) using only the variable t, not x)
2. Express \(\mathrm{4^x}\) using base 2: Since we're given information about powers of 2, rewrite \(\mathrm{4^x}\) using base 2 to create a common foundation
3. Manipulate the given equation to isolate useful components: Rearrange \(\mathrm{t = 2^{(x+1)}}\) to find expressions involving \(\mathrm{2^x}\) that we can substitute
4. Substitute and simplify to express in terms of t: Use the relationship found in the previous step to replace components in our \(\mathrm{4^x}\) expression with t

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're being asked to do. We have an equation that tells us \(\mathrm{t = 2^{(x+1)}}\), and we want to find what \(\mathrm{4^x}\) equals when we write it in terms of t only (no x should appear in our final answer).

Think of it like this: we have two different exponential expressions involving the same variable x, and we want to connect them through the variable t. It's like having two different recipes that use the same ingredient, and we want to figure out how much of one recipe we need based on how much we made of the other recipe.

Process Skill: TRANSLATE

2. Express \(\mathrm{4^x}\) using base 2

Since our given information involves powers of 2 (because \(\mathrm{t = 2^{(x+1)}}\)), let's rewrite \(\mathrm{4^x}\) using base 2 so everything uses the same "language."

What is 4 in terms of powers of 2? Well, \(\mathrm{4 = 2^2}\).

So \(\mathrm{4^x = (2^2)^x}\)

Using the power rule for exponents (when you raise a power to a power, you multiply the exponents): \(\mathrm{4^x = (2^2)^x = 2^{(2x)}}\)

Now we have \(\mathrm{4^x = 2^{(2x)}}\), which is expressed entirely in terms of base 2, just like our given equation.

3. Manipulate the given equation to isolate useful components

We know that \(\mathrm{t = 2^{(x+1)}}\). Let's see if we can find a way to get \(\mathrm{2^x}\) by itself, since that might help us work with \(\mathrm{2^{(2x)}}\).

Starting with \(\mathrm{t = 2^{(x+1)}}\), we can rewrite the exponent: \(\mathrm{t = 2^{(x+1)} = 2^x \cdot 2^1 = 2^x \cdot 2}\)

So: \(\mathrm{t = 2 \cdot 2^x}\)

Dividing both sides by 2: \(\mathrm{t/2 = 2^x}\)

This gives us: \(\mathrm{2^x = t/2}\)

This is perfect! We now have \(\mathrm{2^x}\) expressed in terms of t.

Process Skill: MANIPULATE

4. Substitute and simplify to express in terms of t

Now we can use our result from step 3 to find \(\mathrm{4^x}\) in terms of t.

From step 2, we found: \(\mathrm{4^x = 2^{(2x)}}\) We can rewrite \(\mathrm{2^{(2x)}}\) as: \(\mathrm{2^{(2x)} = (2^x)^2}\)

From step 3, we know: \(\mathrm{2^x = t/2}\)

Substituting this into our expression: \(\mathrm{4^x = (2^x)^2 = (t/2)^2 = t^2/4}\)

Therefore: \(\mathrm{4^x = t^2/4}\)

5. Final Answer

We found that \(\mathrm{4^x = t^2/4}\).

Looking at our answer choices:

1. t
2. t/2
3. t²
4. t²/2
5. t²/4

Our answer matches choice E: \(\mathrm{t^2/4}\).

To verify: We started with \(\mathrm{t = 2^{(x+1)}}\), found that \(\mathrm{2^x = t/2}\), and used this to show that \(\mathrm{4^x = 2^{(2x)} = (2^x)^2 = (t/2)^2 = t^2/4}\). ✓

Common Faltering Points

Errors while devising the approach

1. Attempting to solve for x directly instead of expressing in terms of t

Students often miss that the question asks for \(\mathrm{4^x}\) "in terms of t" and instead try to find the numerical value of x first. They might attempt to solve \(\mathrm{t = 2^{(x+1)}}\) by taking logarithms to find x, then substitute back. This approach is unnecessarily complex and misses the elegant algebraic manipulation required.

2. Not recognizing the need to use the same base

Many students fail to see that since the given equation involves base 2 (\(\mathrm{t = 2^{(x+1)}}\)), they should express \(\mathrm{4^x}\) using base 2 as well. They might try to work with \(\mathrm{4^x}\) directly without converting 4 to \(\mathrm{2^2}\), making the connection between the expressions much more difficult to establish.

Errors while executing the approach

1. Incorrect application of exponent rules

When converting \(\mathrm{4^x}\) to base 2, students often make errors with the power rule. They might write \(\mathrm{4^x = (2^2)^x = 2^{(2+x)}}\) instead of the correct \(\mathrm{2^{(2x)}}\), confusing addition and multiplication of exponents.

2. Algebraic manipulation errors when isolating \(\mathrm{2^x}\)

Starting from \(\mathrm{t = 2^{(x+1)} = 2 \cdot 2^x}\), students frequently make errors when solving for \(\mathrm{2^x}\). They might incorrectly write \(\mathrm{2^x = t + 2}\) or \(\mathrm{2^x = 2t}\) instead of the correct \(\mathrm{2^x = t/2}\), confusing the inverse operation needed.

3. Squaring fractions incorrectly

When computing \(\mathrm{(t/2)^2}\), students often make the error of writing \(\mathrm{t^2/2}\) instead of the correct \(\mathrm{t^2/4}\). They forget that when squaring a fraction, both the numerator AND denominator must be squared: \(\mathrm{(t/2)^2 = t^2/2^2 = t^2/4}\).

Errors while selecting the answer

No likely faltering points - the final algebraic result clearly matches one of the given answer choices, making selection straightforward once the calculation is complete.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient value for x

Let's choose \(\mathrm{x = 3}\), which makes the calculations clean and avoids fractions.

Step 2: Calculate the corresponding value of t

Given: \(\mathrm{t = 2^{(x+1)}}\)

With \(\mathrm{x = 3}\): \(\mathrm{t = 2^{(3+1)} = 2^4 = 16}\)

Step 3: Calculate \(\mathrm{4^x}\) with our chosen value

\(\mathrm{4^x = 4^3 = 64}\)

Step 4: Test each answer choice to see which equals 64

Since \(\mathrm{t = 16}\):

• \(\mathrm{t = 16 \neq 64}\)
• \(\mathrm{t/2 = 16/2 = 8 \neq 64}\)
• \(\mathrm{t^2 = 16^2 = 256 \neq 64}\)
• \(\mathrm{t^2/2 = 256/2 = 128 \neq 64}\)
• \(\mathrm{t^2/4 = 256/4 = 64}\) ✓

Step 5: Verify with a second value

Let's try \(\mathrm{x = 1}\) to double-check:

• \(\mathrm{t = 2^{(1+1)} = 2^2 = 4}\)
• \(\mathrm{4^x = 4^1 = 4}\)
• \(\mathrm{t^2/4 = 4^2/4 = 16/4 = 4}\) ✓

The answer is consistently E across different values, confirming our result.