Question

\(2^x + 2^x =\)

• A. \(2^{x+1}\)
• B. \(2^{x+2}\)
• C. \(2^{2x}\)
• D. \(4^x\)
• E. \(4^{2x}\)

Hint:

A common mistake with exponents is to add them during addition (e.g., thinking \(2^x + 2^x = 2^{2x}\)). This is incorrect. Exponents are added during multiplication of like bases. Always remember to factor first when adding identical exponential terms.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem tests the rules of exponents and algebraic simplification. We are adding two identical terms, which can be simplified through factoring.
Step 2: Key Formula or Approach:
The key exponent rule is \(a^m \times a^n = a^{m+n}\).
The algebraic approach is to factor out the common term.
Step 3: Detailed Explanation:
We are given the expression \(2^x + 2^x\).
Think of \(2^x\) as a single variable, like \(y\). The expression is equivalent to \(y + y\), which simplifies to \(2y\).
Substituting \(2^x\) back in for \(y\), we get:
\[ 2 \times (2^x) \]
This can be written as:
\[ 2^1 \times 2^x \]
Now, using the exponent rule \(a^m \times a^n = a^{m+n}\), we add the exponents:
\[ 2^{1+x} \text{ or } 2^{x+1} \]
Let's check the options. \(2^{x+1}\) matches option (A). Note that \(4^x = (2^2)^x = 2^{2x}\), which is different.
Step 4: Final Answer:
The expression \(2^x + 2^x\) simplifies to \(2^{x+1}\).