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:

📰 Why Asheville Reasonably Relies on Septa Rail Like Never Before! 📰 Septa Regional Rail Schedule Leak: The Game-Changing Timing Everyone Needs 📰 Septa Regional Rail Just Shook Philadelphia’s Commute—Here’s What Officials Won’t Say 📰 View This Boyyoull Never Look The Same Again 684816 📰 How Many Seasons Of Landman 8443951 📰 Gracie Abrams Abs 2928920 📰 Spam Nutrition Facts 6120793 📰 Is El Pollo Stock The Next Viral Food Sensation Authority Proves Its A Game Changer 7658746 📰 Barbarian Pathfinder 1486600 📰 Shocked By The Flavor Ditalini Pasta Dweals You In 3 Seconds 6116578 📰 Scorpio Taurus Romance The Ultimate Power Pairing That Sparks Passion Secrets Inside 5177519 📰 Your Soul Is Wakingdont Miss The Quiet Power Of Morning Prayer 461723 📰 Campintouch 7370628 📰 Jason Without Mask The Dramatic Reveal Everyones Been Waiting Foryou Wont Look Away 4100000 📰 These Landscape Forms Will Change How You See Nature Foreversee Them Before Its Too Late 1026214 📰 Wellsfargfo 3146511 📰 Stop Waitingbank Like Never Before At Your Most Convenient Location 6241872 📰 Mortal Kombat 1 Arcade Moves 1186988

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