Solving $10^x imes 5^{(2x-2)} imes 4^{(x-1)} = 1$: A Math Guide

Hey guys! Let's dive into solving this interesting exponential equation: $10^x imes 5^{(2x-2)} imes 4^{(x-1)} = 1$. If you're scratching your head, don't worry! We're going to break it down step by step. Exponential equations might seem daunting at first, but with the right approach, they can be quite manageable. Our goal is to find the value of x that makes this equation true. So, grab your thinking caps, and let's get started!

Understanding the Equation

First, let's understand the equation we're dealing with: $10^x imes 5^{(2x-2)} imes 4^{(x-1)} = 1$. This equation involves exponential terms, which means the variable x is in the exponent. To solve it, we need to manipulate the equation using the properties of exponents and logarithms until we isolate x. This usually involves simplifying the expressions and finding common bases if possible. The key here is to remember the fundamental rules of exponents, such as $a^{m+n} = a^m imes a^n$ and $(a^m)^n = a^{mn}$. By applying these rules strategically, we can transform the equation into a more solvable form. Keep in mind that our ultimate goal is to get x by itself on one side of the equation, so each step we take should bring us closer to that goal. Sometimes, it helps to rewrite numbers in terms of their prime factors, as this can reveal hidden simplifications. For instance, we can express 10 as 2 × 5 and 4 as $2^2$, which might lead to further simplifications when plugged back into the original equation.

Step-by-Step Solution

Now, let's get into the step-by-step solution. This is where we'll actually start manipulating the equation to find the value of x. Ready? Here we go:

1. Rewrite the equation: Our initial equation is $10^x imes 5^{(2x-2)} imes 4^{(x-1)} = 1$. The first thing we'll do is rewrite the terms to have prime factor bases. Remember that $10 = 2 imes 5$ and $4 = 2^2$. Substituting these into the equation, we get: $(2 imes 5)^x imes 5^{(2x-2)} imes (2^2)^{(x-1)} = 1$

2. Apply exponent rules: Next, we'll use the rule $(ab)^n = a^n imes b^n$ and $(a^m)^n = a^{mn}$ to simplify the equation further: $2^x imes 5^x imes 5^{(2x-2)} imes 2^{2(x-1)} = 1$ $2^x imes 5^x imes 5^{(2x-2)} imes 2^{(2x-2)} = 1$

3. Combine like bases: Now, let's group the terms with the same base together. We'll combine the terms with base 2 and the terms with base 5 using the rule $a^m imes a^n = a^{m+n}$: $2^{(x + 2x - 2)} imes 5^{(x + 2x - 2)} = 1$ $2^{(3x - 2)} imes 5^{(3x - 2)} = 1$

4. Rewrite the equation again: Notice that both terms now have the same exponent, $(3x - 2)$. We can rewrite the equation using the rule $a^n imes b^n = (ab)^n$: $(2 imes 5)^{(3x - 2)} = 1$ $10^{(3x - 2)} = 1$

5. Solve for x: To solve for x, we need to remember that any non-zero number raised to the power of 0 is equal to 1. So, $10^0 = 1$. This means that the exponent $(3x - 2)$ must be equal to 0: $3x - 2 = 0$ Now, solve for x: $3x = 2$ x = rac{2}{3}

Verifying the Solution

It's always a good idea to verify the solution to make sure we didn't make any mistakes along the way. We'll plug x = rac{2}{3} back into the original equation:

10^{(rac{2}{3})} imes 5^{(2(rac{2}{3})-2)} imes 4^{(rac{2}{3}-1)} = 1

Let's simplify each part:

• 10^{(rac{2}{3})} remains as 10^{(rac{2}{3})}
• 5^{(2(rac{2}{3})-2)} = 5^{(rac{4}{3}-2)} = 5^{(rac{4}{3}-rac{6}{3})} = 5^{(-rac{2}{3})}
• 4^{(rac{2}{3}-1)} = 4^{(-rac{1}{3})}

So, the equation becomes:

10^{(rac{2}{3})} imes 5^{(-rac{2}{3})} imes 4^{(-rac{1}{3})} = 1

Rewrite 4 as $2^2$:

10^{(rac{2}{3})} imes 5^{(-rac{2}{3})} imes (2^2)^{(-rac{1}{3})} = 1

10^{(rac{2}{3})} imes 5^{(-rac{2}{3})} imes 2^{(-rac{2}{3})} = 1

Rewrite 10 as $2 imes 5$:

(2 imes 5)^{(rac{2}{3})} imes 5^{(-rac{2}{3})} imes 2^{(-rac{2}{3})} = 1

2^{(rac{2}{3})} imes 5^{(rac{2}{3})} imes 5^{(-rac{2}{3})} imes 2^{(-rac{2}{3})} = 1

Combine the terms:

2^{(rac{2}{3}-rac{2}{3})} imes 5^{(rac{2}{3}-rac{2}{3})} = 1

$2^0 imes 5^0 = 1$

$1 imes 1 = 1$

$1 = 1$

Our solution checks out! This verification step is crucial because it confirms that our algebraic manipulations were correct and that the value we found for x indeed satisfies the original equation. By plugging the solution back into the initial equation and simplifying, we ensure that both sides of the equation are equal, which validates our answer.

Common Mistakes to Avoid

When solving exponential equations, there are some common mistakes to avoid. Recognizing these pitfalls can save you a lot of time and frustration. One frequent error is misapplying exponent rules. For example, students sometimes incorrectly distribute exponents over sums or differences. Remember, $(a + b)^n$ is not the same as $a^n + b^n$. Another common mistake is failing to correctly combine like terms. When you have terms with the same base, make sure you add or subtract the exponents properly. A third mistake is forgetting to check your solution. As we demonstrated earlier, plugging your solution back into the original equation is crucial to verify its correctness. If the solution doesn't satisfy the original equation, it's likely that a mistake was made in the solving process. Finally, be careful with negative exponents. A negative exponent indicates a reciprocal, so a^{-n} = rac{1}{a^n}. Keeping these common mistakes in mind and double-checking your work will greatly improve your accuracy in solving exponential equations.

Conclusion

Alright, guys, we've successfully solved the equation $10^x imes 5^{(2x-2)} imes 4^{(x-1)} = 1$, and we found that x = rac{2}{3}. Remember, the key to solving exponential equations is to simplify, use the exponent rules correctly, and always verify your solution. Keep practicing, and you'll become a pro at these in no time! Solving exponential equations can seem like a complex task, but by breaking it down into manageable steps and applying the fundamental rules of exponents, you can tackle even the most challenging problems. The journey we've taken today illustrates the importance of careful manipulation, strategic simplification, and thorough verification. Each step, from rewriting the equation with prime factor bases to combining like terms and solving for x, builds upon the previous one, leading us to the final solution. So, keep honing your skills, stay patient, and remember that with practice, you'll become more confident and proficient in solving exponential equations. Until next time, happy solving!