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.

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]

Image Gallery

$algebra precalculus - Calculate $\frac {1}{2\cdot3\cdot 4}+\frac {1}{3 ...$

$Solved Show that \\( -3 \\cdot \\log (x)+6 \\cdot x^{2}+6 | Chegg.com$

$x+x \cdot a \cdot x-x \cdot a+x-a-a \cdot a=8 \\|x|^{2}-\left.1 a\right|..$

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

🔗 Related Articles You Might Like:

📰 Bank of America Visa Customer Service Number 📰 Bank of America Branford 📰 Bank of America Card Dispute 📰 You Wont Believe How Bright The Air Max 95 Neon Shootsshock Yourclients Style Now 1297404 📰 Unlock The Secrets Of Texas Holdem The Cheat Sheet Thats Taking Online Pokies By Storm 1867929 📰 Dont Miss The Market Openheres The Exact Nasdaq Opening Time Every Trader Must Know 6685053 📰 The Ultimate Breakdown How Much Do Weddings Cost Prepare To Be Shocked By Hidden Fees 1901065 📰 No Internet Still Need Visio 2019 Grab The Offline Installer Now 5942247 📰 Where To Stay In Rome 6146730 📰 Washington Dc Protest 4659006 📰 Best Golf Drivers 2025 5051952 📰 Cate School 5732319 📰 Becky Weiss 6705498 📰 Winning Powerball Numbers Last Night 1349335 📰 Serum Free Download 4010191 📰 Cast Of The Perfect Setting 4143724 📰 C Baldwin Hotel Houston 5422798 📰 Firestick 9974221

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