The Question
Solve for x: 2^(x+4)+2^x = 8704
Details
💡 Hint
Look for ways to simplify the terms with ‘x’ in the exponent. Can you use exponent rules to create a common factor?
📝 Solution Steps
- Apply exponent rules to expand terms like 2^(x+4).
- Factor out the common exponential term.
- Simplify the numerical part of the equation.
- Isolate the exponential term (e.g., 2^x).
- Express the constant on the right side as a power of the same base as the exponential term.
- Equate the exponents to solve for x.
📚 Explanation
This is an exponential equation. The trick here is to use exponent rules to simplify the left side. Remember that a^(m+n) = a^m * a^n. So, 2^(x+4) can be rewritten as 2^x * 2^4. Now the equation becomes 2^x * 2^4 + 2^x = 8704. You can factor out the common term 2^x: 2^x * (2^4 + 1) = 8704. Calculate 2^4, which is 16. So, 2^x * (16 + 1) = 8704, which simplifies to 2^x * 17 = 8704. Divide both sides by 17 to get 2^x = 512. Finally, express 512 as a power of 2 (512 = 2^9). Since the bases are the same, you can equate the exponents: x=9.
✅ Answer
x = 9
⚠️ Common Mistakes
- Incorrect application of exponent rules (e.g., adding exponents when multiplying bases).
- Calculation errors when simplifying numerical terms.
📐 Formulas Required
- a^(m+n) = a^m * a^n
