Solving the Equation: Unraveling the Mystery of x in (12^𝑥+12^𝑥+12^𝑥)/(6^𝑥+6^𝑥)=48

Solving the Equation: Unraveling the Mystery of x in (12^𝑥+12^𝑥+12^𝑥)/(6^𝑥+6^𝑥)=48

#manojagarwal #empoweringwisdomchannel #mathsolympiad

Introduction

In the world of mathematics, equations often serve as fascinating puzzles, waiting to be solved. In this blog post, we’ll explore one such puzzle—a seemingly intricate equation that involves exponents and ratios. The equation in question is:

(12^𝑥+12^𝑥+12^𝑥)/(6^𝑥+6^𝑥) = 48

At first glance, this equation may appear daunting, but with a step-by-step approach and an understanding of mathematical principles, we’ll unveil the mystery of x.

Understanding the Equation

The equation (12^𝑥+12^𝑥+12^𝑥)/(6^𝑥+6^𝑥) = 48 involves exponents and ratios. To solve for x, we must break down the equation and apply mathematical principles effectively.

Step 1: Simplifying the Equation

Let’s start by simplifying the equation. Notice that we have 12^𝑥 in the numerator and 6^𝑥 in the denominator. To make the equation more manageable, we can rewrite it as follows:

(3 * 4^𝑥)/(3 * 2^𝑥) = 48

By canceling out the common factors of 3, we get:

(4^𝑥)/(2^𝑥) = 48

Step 2: Utilizing Exponent Rules

Now that we have the equation in a simpler form, let’s use exponent rules. Specifically, we can apply the rule that states (a^m)/(a^n) = a^(m-n). In this case, a is 4, and m is 𝑥, while a is 2, and n is 𝑥:

4^𝑥 – 2^𝑥 = 48

Step 3: Transforming the Equation

To further simplify, we can reframe the equation:

4^𝑥 – 2^𝑥 = 48

As 2^𝑥 is common to both sides, we can isolate it on one side:

4^𝑥 = 2^𝑥 + 48

Step 4: Expanding 4^𝑥

We can represent 4 as 2^2 and expand 4^𝑥 accordingly:

(2^2)^𝑥 = 2^𝑥 + 48

Using the rule (a^m)^n = a^(m*n), we get:

2^(2𝑥) = 2^𝑥 + 48

Step 5: Equating the Bases

To solve for x, we need to equate the bases, as they are both 2. This implies that the exponents must be equal:

2𝑥 = 𝑥 + 48

Step 6: Solving for x

Subtracting 𝑥 from both sides of the equation:

2𝑥 – 𝑥 = 48

𝑥 = 48

Conclusion

In this blog post, we unraveled the mystery of x in the equation (12^𝑥+12^𝑥+12^𝑥)/(6^𝑥+6^𝑥) = 48. By simplifying the equation, applying exponent rules, and isolating the terms, we successfully solved for x, which equals 48.

Mathematics often presents challenges that require careful thought and systematic approaches. This equation serves as a prime example of how mathematical principles and rules can help us tackle complex problems. Whether you’re a math enthusiast or just curious about equations, this journey to find the value of x demonstrates the power of mathematical reasoning and problem-solving.