2^x \cdot (2^3)^{x-1} = 2^6
Understanding 2^x · (2³)^{x−1} = 2⁶: Solving the Exponential Equation
When faced with the equation 2^x · (2³)^{x−1} = 2⁶, many students and learners wonder how to simplify and solve it efficiently. This problem beautifully demonstrates key principles of exponential expressions, especially the rules of exponents—leveraging powers of powers and product rules—making it a perfect example for practicing algebraic and logarithmic thinking in everyday math and education.
What does 2^x · (2³)^{x−1} = 2⁶ mean?
At first glance, this equation involves exponential terms with the same base—2—so simplifying it comes down to applying essential exponent rules:
- Power of a power: (a^m)^n = a^{m·n}
- Product of powers: a^m · a^n = a^{m+n}
Step-by-Step Simplification
Start with the original equation:
2^x · (2³)^{x−1} = 2⁶
Use the power of a power rule inside the parentheses:
2^x · [2^{3·(x−1)}] = 2⁶
Now apply the product rule:
2^{x + 3(x−1)} = 2⁶
Simplify the exponent on the left:
x + 3(x − 1) = x + 3x − 3 = 4x − 3
So the equation becomes:
2^{4x−3} = 2⁶
Since the bases are equal, set the exponents equal:
4x − 3 = 6
Solve for x:
4x = 6 + 3 = 9 x = 9/4
Final Answer
x = 9⁄4
Why This Equation Matters in Math and Study
This problem isn’t just about solving for a single variable—it’s a concise illustration of applying exponent rules effectively. Understanding how to manipulate exponents like 2^{4x−3} and relate them to a fixed base is foundational in algebra, calculus, and advanced topics like logarithms and exponential growth models.
Key Takeaways
- Exponents with the same base follow straightforward rules: products become sums, powers become multipliers.
- Simplify expressions by combining exponents using algebraic rules.
- Setting equivalent exponents equal gives a direct path to solving exponential equations.
Practice It!
Try solving similar equations:
- 2^x · (2⁴)^{x−2} = 2⁸
- 3^{x+1} · 3^{2x−5} = 3⁶
Use exponent rules to simplify, then set exponents equal.
Mastering exponents like this puts you ahead in algebra and opens the door to more complex mathematical problem-solving every day.
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