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:

📰 Skyrim Khajiit 📰 Harvest Moon Friends of Mineral Town 📰 Big Show Height 📰 Master The Art Of Good Poker Hands These Cards Dominate Every Table 6829715 📰 Wells Fargo Prattville Al 📰 Live Update X Men Characters Men And The Fallout Continues 📰 Ilzze Ozza Exposed The Truth That Opened My Eyes Forever 5489193 📰 Comparing Car Insurance 📰 Wells Fargo Truck Payment 📰 Ny Manhattan Zip Code Breakdown Profile Of The Most Savage Eye Opening Neighborhoods 4922555 📰 Stages Of Economic Growth By Rostow 4992640 📰 Verizon Com Expresspay 📰 Unlock Your Files In Minutes Microsoft Bitlocker Key Recovery Guide 8428499 📰 Download Free Macintosh Games 5518804 📰 Irtv24 Exposes Secrets No One Dares To Mention 94966 📰 Bob Discount 2823590 📰 Transmision Torrent 📰 The Ultimate List Of Best Two Player Games For Endless Fun 2629101

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