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.

Understanding the Context

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}
Solved: 2 x ·(x+1)+3 x=3 ·(x+1)^2 [Math]

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Solved: 2 x ·(x+1)+3 x=3 ·(x+1)^2 [Math]
algebra precalculus - Calculate $\frac {1}{2\cdot3\cdot 4}+\frac {1}{3 ...
Solved Show that \\( -3 \\cdot \\log (x)+6 \\cdot x^{2}+6 | Chegg.com
x+x \cdot a \cdot x-x \cdot a+x-a-a \cdot a=8 \\|x|^{2}-\left.1 a\right|..
x+x \cdot a \cdot x-x \cdot a+x-a-a \cdot a=8 \\|x|^{2}-\left.1 a\right|..

Key Insights

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

Continue Reading

For the 20th term, $a_{20} = 3(20) - 1 = 59$. However, the question asks for the count of terms, not the value. Since the sequence starts at $n = 1$ and ends at $n = 20$, there are $\boxed{20}$ numbers.Question: A square has a perimeter of $ 40 \, \text{cm} $. If each side is increased by $ 2 \, \text{cm} $, by how many square centimeters does the area increase?
Solution: The original side length of the square is $ \frac{40}{4} = 10 \, \text{cm} $. The original area is $ 10^2 = 100 \, \text{cm}^2 $. After increasing each side by $ 2 \, \text{cm} $, the new side length is $ 12 \, \text{cm} $, and the new area is $ 12^2 = 144 \, \text{cm}^2 $. The increase in area is $ 144 - 100 = 44 \, \text{cm}^2 $.
\boxed{44}

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Final Thoughts

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