Unlocking Exponents: Which Equations Hold True?
Hey math enthusiasts! Let's dive into the fascinating world of exponents. In this article, we'll break down the rules of exponents and figure out which equations hold true. This is important stuff, so pay close attention. It's all about understanding how exponents work when you multiply terms. So, buckle up, grab your pencils, and let's get started. We'll be looking at several equations and determining whether they are accurate. Remember, the key to mastering exponents is practice, so we will walk through each step so you can easily understand.
The Core Concept of Exponents
First, let's refresh our memory on the fundamental principle of exponents. When multiplying exponential terms with the same base, you add the exponents. In mathematical terms, this is represented as: $x^m imes x^n = x^{m+n}$. Where 'x' is the base, and 'm' and 'n' are the exponents. This rule is super important, so try to remember it. So, basically, you keep the base the same and add the powers together. This is the golden rule! This simple rule unlocks a whole world of possibilities when simplifying and solving exponential equations. This rule is your best friend when dealing with exponents. When you understand this, you're on the right track. This concept is fundamental to solving problems involving exponents.
So, with this in mind, we can start to assess the options and see which ones are correct. Remember, the goal is to find the equations that correctly apply the rule of adding exponents when multiplying terms with the same base. Let's start with the first option and see if it is correct or not. Always be on the lookout to apply this when dealing with exponents. This skill is useful in higher-level math and real-world applications. Practice these steps and you'll be a pro in no time.
Analyzing the Equations
Now, let's analyze each equation one by one to determine if it's true. Remember, the goal is to find the equations that correctly follow the rules of exponents. We need to check if the exponents are added correctly. Remember the golden rule: $x^m imes x^n = x^m+n}$. Let's start with option A. Does it follow the rule? Let's take a look at option A first. In equation A, we have imes x^4 = x^{48}$. According to the rule of exponents, when multiplying terms with the same base, we add the exponents. So, we should add 12 and 4. 12 plus 4 equals 16, not 48. Therefore, option A is false. The correct equation should be $x^{12} imes x^4 = x^{16}$. See, it's not that hard, right?
Now, let's check option B. Option B states that: $x^4 imes x^{12} = x^{48}$. Again, according to the rule, we need to add the exponents. Adding 4 and 12, we get 16. So, the correct result should be $x^{16}$, not $x^{48}$. Thus, option B is also not correct. Option B is similar to Option A, but the order of the terms is different. But the principle of exponents still applies here. Remember, practice is super important, so try to do this yourself. You will improve every time you do it.
Next, let's move on to option C. Option C says: $x^{11} imes x^{37} = x^{48}$. To verify this, we should add the exponents 11 and 37. Adding them together gives us 48. So, the right side of the equation is also 48. Thus, this equation is true. So, this looks like a correct answer. Hooray, we got one! See, it's not that hard, right? Keep practicing the rule and you will ace it. Remember the rule, keep the base and add the exponents. You got this!
Let's keep going and check option D. Option D is: $x^0 imes x^{48} = x^{48}$. Remember that any number raised to the power of 0 is equal to 1. That means $x^0 = 1$. Therefore, the left side of the equation simplifies to $1 imes x^{48}$. This equals to $x^{48}$, so option D is indeed true. Great, another correct answer! Easy, peasy! Remember, anything to the power of zero is one.
Finally, we will examine option E. Option E is: $x^{24} imes x^{2} = x^{48}$. Let's add the exponents, 24 and 2. The result is 26, not 48. So, this option is false. The correct answer should be $x^{24} imes x^{2} = x^{26}$. This is not correct. But keep practicing this to improve your skills. Now that we have analyzed all options, let's summarise our findings. Always remember the rules and you will do great. You are doing great!
Identifying the Correct Answers
After examining all the equations, let's identify which ones are true. Remember that the rule to use is $x^m imes x^n = x^m+n}$. Based on our analysis imes x^37} = x^{48}$, is true, as 11 plus 37 equals 48. Option D = x^{48}$, is also true, because $x^0 = 1$, and $1 imes x^{48} = x^{48}$. So, the correct answers are C and D. Congratulations, you did it! Always go step by step to find the correct answer. You can do it! Keep going, you are doing great.
Conclusion
And there you have it, guys! We have successfully navigated through the world of exponents and identified the true equations. By understanding and applying the fundamental rule of exponents, we could easily determine the correct answers. Remember, the key is to keep practicing and always double-check your calculations. Keep in mind the golden rule: when multiplying exponential terms with the same base, you add the exponents. Now, you are ready to use your skills in more complex math problems. Keep practicing and you will be a math expert soon! If you have any further questions, feel free to ask. That's all for today, folks! Keep practicing and have fun with math.