Solve: 4096 x = 8
The correct answer and explanation is:
We are given the equation: 4096x=84096^x = 8
Step 1: Express both sides as powers of 2
To solve for xx, it’s helpful to express both 4096 and 8 as powers of the same base.
We know:
- 212=40962^12 = 4096
- 23=82^3 = 8
So we can rewrite the equation using base 2: (212)x=23(2^{12})^x = 2^3
Step 2: Apply the rule of exponents
Using the exponent rule (am)n=am⋅n(a^m)^n = a^{m \cdot n}, we get: 212x=232^{12x} = 2^3
Since the bases are the same, we can set the exponents equal to each other: 12x=312x = 3
Step 3: Solve for xx
x=312=14x = \frac{3}{12} = \frac{1}{4}
✅ Final Answer:
x=14x = \frac{1}{4}
📘 Explanation (300 words):
This problem involves solving an exponential equation, where the variable xx is in the exponent. The key to solving such equations is to express both sides using the same base.
First, observe the numbers. The number 4096 might seem large, but it’s a power of 2. In fact: 212=40962^{12} = 4096
Similarly, 8 is a small and well-known power of 2: 23=82^3 = 8
By writing both sides of the equation with base 2, we get: (212)x=23(2^{12})^x = 2^3
We then use the rule of exponents that states: (am)n=am⋅n(a^m)^n = a^{m \cdot n}. Applying that here simplifies the left-hand side: 212x=232^{12x} = 2^3
Now that both sides have the same base (2), the only way the equation can be true is if the exponents are equal: 12x=312x = 3
Solving for xx, we divide both sides by 12: x=312=14x = \frac{3}{12} = \frac{1}{4}
This means that raising 4096 to the power of 14\frac{1}{4} equals 8. That checks out because taking the fourth root of 4096 gives 8.
So, the final answer is: x=14x = \frac{1}{4}