Solving For T In The Exponential Equation 9^7 / ⁴√9^10 = 9^(6t)

Hey there, math enthusiasts! Ever stumbled upon an equation that looks like it's speaking a different language? Well, today we're going to decode one such equation and find the value of 't' in it. Buckle up, because we're diving into the world of exponents and roots! We'll break down every step, ensuring you not only understand the solution but also grasp the underlying concepts. Let's make math fun and accessible, one equation at a time.

Decoding the Equation: 9⁷ / ⁴√9¹⁰ = 9⁶ᵗ

Alright, let's get straight to the heart of the matter. We're faced with the equation 9⁷ / ⁴√9¹⁰ = 9⁶ᵗ, and our mission, should we choose to accept it (spoiler: we do!), is to find the value of 't'. At first glance, it might seem like a jumble of numbers and symbols, but don't worry, we're going to dissect it piece by piece. The key here is to understand the rules of exponents and how they interact with roots. Remember, mathematics is like a puzzle, and every piece fits perfectly once you know where it belongs. Our journey begins with simplifying the equation, turning complex-looking terms into simpler ones. This is where the magic of exponent rules comes into play. We'll be using rules like a⁻ⁿ = 1/aⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, and (aᵐ)ⁿ = aᵐⁿ. So, keep these in mind as we move forward. The goal is to express both sides of the equation with the same base, which in this case is 9. Once we have the same base on both sides, we can simply equate the exponents and solve for 't'. It's like comparing apples to apples, instead of apples to oranges – a much easier task, wouldn't you agree? So, let's roll up our sleeves and get to work on this fascinating mathematical puzzle.

Step 1: Simplifying the Root

Our initial focus is on tackling the root in the denominator: ⁴√9¹⁰. Roots can sometimes look intimidating, but they're really just fractional exponents in disguise. Think of it this way: the nth root of a number is the same as raising that number to the power of 1/n. So, ⁴√x is the same as x¹/⁴. Applying this knowledge to our equation, we can rewrite ⁴√9¹⁰ as (9¹⁰)¹/⁴. Now, we can use another exponent rule that states (aᵐ)ⁿ = aᵐⁿ. This means we multiply the exponents. So, (9¹⁰)¹/⁴ becomes 9¹⁰ˣ¹/⁴, which simplifies to 9¹⁰/⁴ or 9⁵/². See? We've already made significant progress in simplifying the equation. By converting the root into a fractional exponent and applying the power of a power rule, we've transformed a complex term into a more manageable one. This is a crucial step because it allows us to combine this term with the other terms in the equation, which all have exponents. Remember, the key to solving complex problems is often breaking them down into smaller, more manageable steps. And that's exactly what we're doing here. We're taking the equation piece by piece, simplifying each part, and then putting it all back together to find the solution.

Step 2: Rewriting the Equation

Now that we've simplified the root, let's rewrite the entire equation with our new understanding. Originally, we had 9⁷ / ⁴√9¹⁰ = 9⁶ᵗ. We've figured out that ⁴√9¹⁰ is the same as 9⁵/². So, we can substitute that into our equation, giving us 9⁷ / 9⁵/² = 9⁶ᵗ. This is a much cleaner equation already! It's like decluttering your room – once you get rid of the unnecessary stuff, you can see things much more clearly. In this case, we've decluttered the equation by simplifying the root and expressing it as a power of 9. But we're not done yet. We still have a division of exponents to deal with on the left side of the equation. Remember the rule aᵐ/aⁿ = aᵐ⁻ⁿ? This rule is going to be our best friend here. It tells us that when we divide numbers with the same base, we can simply subtract their exponents. So, 9⁷ / 9⁵/² can be simplified further. We're on the verge of making the equation even simpler and bringing ourselves closer to finding the elusive value of 't'. Keep following along, and you'll see how these seemingly complex equations can be tamed with the right tools and techniques. The beauty of mathematics lies in its consistency – the rules always apply, and once you understand them, you can solve almost any problem.

Step 3: Applying the Quotient Rule

Alright, let's put that quotient rule to work! We have 9⁷ / 9⁵/², and as we discussed, the rule aᵐ/aⁿ = aᵐ⁻ⁿ tells us to subtract the exponents. So, we need to calculate 7 - 5/2. To do this, we need a common denominator. We can rewrite 7 as 14/2. Now we have 14/2 - 5/2, which equals 9/2. This means 9⁷ / 9⁵/² simplifies to 9⁹/². Wow, we're making some serious progress! The left side of the equation is looking much simpler now. By applying the quotient rule, we've combined the two terms into a single term with a single exponent. This is a key step in solving exponential equations – getting both sides of the equation into a similar form so we can compare them. It's like translating two different languages into a common language so you can understand the message. Now our equation looks like this: 9⁹/² = 9⁶ᵗ. The finish line is in sight! We're just one step away from finding the value of 't'. Keep your focus, and let's bring this equation home.

Step 4: Equating the Exponents

Here comes the pivotal moment! We've successfully simplified our equation to 9⁹/² = 9⁶ᵗ. Now, we have the same base (which is 9) on both sides of the equation. This is where the magic happens. When the bases are the same, we can simply equate the exponents. It's like saying if 2ˣ = 2ʸ, then x must equal y. In our case, this means 9/2 = 6t. We've transformed an exponential equation into a simple algebraic equation. The heavy lifting is done, and now it's just a matter of solving for 't'. This step highlights the power of simplification in mathematics. By using the rules of exponents and roots, we were able to reduce a complex equation into a straightforward one. Now, we're just one simple algebraic manipulation away from finding our answer. Get ready to unveil the value of 't', because we're about to crack the code!

Step 5: Solving for t

We're in the home stretch now! We have the equation 9/2 = 6t. To solve for 't', we need to isolate it on one side of the equation. The way we do that is by dividing both sides of the equation by 6. This gives us (9/2) / 6 = t. Now, dividing by 6 is the same as multiplying by 1/6, so we have (9/2) * (1/6) = t. Multiplying the fractions, we get 9/12 = t. But we're not quite done yet. We can simplify the fraction 9/12 by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This gives us 3/4 = t. And there we have it! We've found the value of t. It's been a journey of simplification, application of exponent rules, and finally, a simple algebraic solution. But the satisfaction of solving a problem like this is what makes mathematics so rewarding. You've taken a complex equation and, step by step, broken it down into its simplest form and found the answer. That's the power of mathematical thinking! So, the value of t is 3/4. Congratulations on solving this exponent mystery!

The Grand Finale: t = 3/4

So, after all our hard work, simplification, and equation-solving prowess, we've arrived at the solution: t = 3/4. Give yourself a pat on the back; you've successfully navigated the world of exponents and roots! We started with a seemingly complex equation, 9⁷ / ⁴√9¹⁰ = 9⁶ᵗ, and through a series of steps, including simplifying the root, applying the quotient rule for exponents, and equating exponents, we've unearthed the value of 't'. This journey highlights the importance of understanding fundamental mathematical principles and how they can be applied to solve problems. It's not just about memorizing formulas; it's about understanding the logic behind them and using them strategically. Remember, mathematics is a building block subject. Each concept builds upon the previous one, and mastering the basics is crucial for tackling more advanced topics. So, keep practicing, keep exploring, and keep challenging yourself. The world of mathematics is vast and fascinating, and there's always something new to learn. And remember, every complex problem can be solved if you break it down into smaller, more manageable steps. You've proven that today by solving for 't'! So, keep up the great work, and who knows what mathematical mysteries you'll unravel next?