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:

📰 albany basketball 📰 northwestern high school 📰 cat flea biting 📰 Flight Seat Map 9357188 📰 Arknet Unleashed The Strange New Futuristic Platform Taking Over The Blockchain Scene 5159799 📰 Osmosis Jones Osmosis 3679133 📰 Frac53 Frac1206 20 5925435 📰 San Palmas 4695060 📰 Never Lose Access Evermaster This Pro Trick That Keeps Login Always Open 9606024 📰 April 18Th Holiday 794142 📰 Dating After 50 These Top Sites Will Change Your Life Forever 5075163 📰 The Shocking Dollar Amount Behind The Biggest Indoor Piercing Price Tag 1907863 📰 How To Make Side Money 8594924 📰 Onsite Vs On Site 2107091 📰 How To Reset Router 5523521 📰 Dodgers Vs Tigers 89553 📰 Texas Chainsaw Madness The Most Brutal Movies You Cant Miss In 2024 1140836 📰 What Are Pennies Worth 4759054

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