Solve For X & Evaluate Exponents: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into a couple of interesting math problems. First up, we'll figure out the value of 'x' in an exponential equation. Then, we'll evaluate an expression involving exponents and fractions. Let's break it down step-by-step, making sure it's super easy to understand. Ready to jump in? Let's go!

Finding the Value of x in 3ˣ + 3⁻¹ = 3¹

Alright, guys, let's tackle our first challenge: finding the value of 'x' in the equation 3ˣ + 3⁻¹ = 3¹. This involves understanding the fundamentals of exponents. Don't worry, it's not as scary as it looks. We'll simplify this equation and isolate 'x' to find its value. This problem tests your understanding of exponential rules and how to manipulate equations. It's a fundamental concept in algebra, so understanding it is crucial. This is going to be fun, let's get started.

Firstly, we have to simplify the equation given. We know that 3¹ is simply 3. Let's also rewrite 3⁻¹ as a fraction. Remember, a negative exponent means we take the reciprocal of the base raised to the positive value of the exponent. So, 3⁻¹ is the same as 1/3. Our equation now looks like this: 3ˣ + (1/3) = 3. Now, the main goal is to isolate 'x.' To start, we need to subtract 1/3 from both sides of the equation. This will leave us with 3ˣ = 3 - (1/3). This is where a little bit of fraction arithmetic comes in handy. You can think of 3 as 3/1, and to subtract 1/3, we need a common denominator, which is 3. So, 3/1 becomes 9/3. Then, we subtract: 9/3 - 1/3 = 8/3. Our equation is now 3ˣ = 8/3. To find 'x,' we need to think about how to get rid of the base 3. The most common way to do this is by using logarithms. To solve for 'x,' take the logarithm of both sides. In this case, we could use the logarithm base 3, but for simplicity, we can use the natural logarithm (ln). So, we have: ln(3ˣ) = ln(8/3). Now, using the power rule of logarithms, we can bring the exponent 'x' down: x * ln(3) = ln(8/3). Finally, to isolate 'x,' we divide both sides by ln(3): x = ln(8/3) / ln(3).

Now, you can use a calculator to find the approximate value of 'x.' Calculate ln(8/3) and ln(3) separately and then divide the first result by the second. This will give you the solution for 'x.' Remember that logarithms are the inverse of exponents, so this approach allows us to solve for an exponent that's part of an equation. Using this method, we can solve various problems using exponents. This method can also be used to solve different kinds of problems, and it’s a foundational skill for more complex math.

Practical Application and Key Takeaways

This type of problem might seem theoretical, but understanding exponents and how to solve equations like this has practical applications in many fields. For example, in finance, exponential functions are used to model compound interest. In computer science, they are used in algorithms and data structures. In physics, they are used to model radioactive decay and growth. Key takeaways are understanding the basic exponent rules, how to manipulate equations to isolate a variable, and how to use logarithms to solve for exponents. Always remember to simplify, isolate, and then solve!

Evaluating (7/10)⁻⁶ * (10/7)⁻²

Alright, friends, let's move on to the second part of our problem: evaluating the expression (7/10)⁻⁶ * (10/7)⁻². This part focuses on evaluating exponents. The expression involves negative exponents and fractions. It's a great exercise to practice your understanding of exponent rules and how they apply to fractions. The main idea here is to manipulate the expression using exponent rules to simplify it and make it easier to calculate the final value. This section will strengthen your grasp of exponential expressions. Let's get started!

First, let's recall the rule for negative exponents. A term raised to a negative exponent is equal to its reciprocal raised to the positive value of the exponent. For instance, a⁻¹ = 1/a. So, we can rewrite (7/10)⁻⁶ as (10/7)⁶. Similarly, we can rewrite (10/7)⁻² as (7/10)². Our expression now becomes (10/7)⁶ * (7/10)². It's often helpful to rewrite fractions. Notice that we have two fractions with different bases, but we can change the base using the negative exponent rule. Now we have an easier expression to evaluate. The best thing to do is to find common denominators.

Next, notice that we can rewrite (10/7)⁶. This means (10/7) multiplied by itself six times. (7/10)² means (7/10) multiplied by itself twice. So we need to simplify this expression. To do this, let's start by rewriting the whole thing. We can rewrite the expression as ((10/7) * (10/7) * (10/7) * (10/7) * (10/7) * (10/7)) * ((7/10) * (7/10)). Now, let's simplify this by canceling out common terms. We have six (10/7) terms multiplied by two (7/10) terms. The key here is realizing that (7/10) and (10/7) are reciprocals. So, for every pair of (10/7) and (7/10), they cancel each other out and simplify to 1. In other words, when you multiply a number by its reciprocal, the result is always 1. So, (10/7) * (7/10) = 1. We have two pairs of these, which cancels out four (10/7) terms. So, we are left with (10/7)⁴. Now, we can evaluate (10/7)⁴ by multiplying (10/7) by itself four times. This is (10 * 10 * 10 * 10) / (7 * 7 * 7 * 7). Which simplifies to 10000/2401. This is the final value of the expression.

Practical Applications and Key Takeaways

This kind of evaluation is fundamental in many areas of mathematics and science. For instance, in physics, it's used to calculate the energy of particles. In engineering, it's useful when dealing with ratios and proportions. Key takeaways here involve understanding how to apply exponent rules to fractions and how to manipulate expressions to make them easier to evaluate. Remember to simplify the expression by rewriting the terms using positive exponents, and always look for opportunities to cancel common factors. Keep practicing, and you'll become more and more comfortable with exponential expressions!

Conclusion: Mastering Exponents and Equations

So, folks, we’ve successfully tackled both problems! We found the value of 'x' in our exponential equation using logarithms and evaluated an expression with negative exponents and fractions. These problems are stepping stones to mastering more complex mathematical concepts. The key is to understand the rules and practice. Remember to break down complex problems into simpler steps, and don’t be afraid to experiment with different approaches.

Keep practicing, keep exploring, and most importantly, have fun with math! There's a whole world of exciting concepts out there waiting for you to discover. If you liked this tutorial, please share it. Let me know what other topics you want to learn about. Thanks for joining me on this math adventure! I hope this step-by-step guide has helped you understand exponents and equations a little bit better. Feel free to reach out if you have any questions or would like to explore more math topics. Happy learning, everyone!