Simplify To 1/n^18: A Math Powers Quest!

by Henrik Larsen 41 views

Hey everyone! Let's dive into an exciting math problem today. We're on a quest to find the expression that simplifies to the captivating form of 1/n^18. This involves understanding the magic of exponents and how they play together. So, grab your thinking caps, and let's get started!

Understanding the Basics of Exponents

Before we jump into the options, let's quickly refresh our memory about the fundamental rules of exponents. These rules are our trusty tools in simplifying complex expressions. Think of them as the secret sauce to solving these kinds of problems.

  • Rule 1: Power of a Power: When you have an exponent raised to another exponent, you multiply them. Mathematically, this looks like (xa)b = x^(a*b). This rule is super important for this problem, so keep it in mind!
  • Rule 2: Negative Exponents: A negative exponent means you take the reciprocal of the base raised to the positive exponent. So, x^(-a) = 1/x^a. This rule is crucial for understanding how to get that 1 in the numerator of our target expression, 1/n^18.
  • Rule 3: Product of Powers: When multiplying exponents with the same base, you add the exponents. This is written as x^a * x^b = x^(a+b). Although not directly used in this particular problem, it's a handy rule to have in your exponent toolkit.
  • Rule 4: Quotient of Powers: When dividing exponents with the same base, you subtract the exponents. This is expressed as x^a / x^b = x^(a-b). Like the previous rule, it's good to know but not central to this specific question.

With these rules in our arsenal, we are now ready to tackle the options and find the one that simplifies to 1/n^18. Let's put our math hats on and start the adventure!

Analyzing Option A: (n2)9

Let's start our investigation with option A: (n2)9. To simplify this expression, we'll use the power of a power rule, which we discussed earlier. Remember, this rule states that when you have an exponent raised to another exponent, you multiply them. In this case, we have n raised to the power of 2, and then that entire term is raised to the power of 9.

Applying the rule, we multiply the exponents 2 and 9. This gives us:

(n2)9 = n^(2*9) = n^18

So, simplifying option A results in n^18. But wait a minute! This isn't quite what we're looking for. We need an expression that simplifies to 1/n^18, and this one gives us n to the positive 18th power. We're close, but not quite there. Option A is like a near miss – it helps us understand the process, but it's not the final answer. We need that negative exponent magic to get the 1 in the numerator, which means we need a negative power of n in our simplified form. Let's move on to the next option and see if it gets us closer to our target.

Deconstructing Option B: (n-9)-2

Next up on our list is option B: (n-9)-2. This expression looks a bit more promising because we see those negative signs! Remember, negative exponents are key to getting a fraction with 1 in the numerator, which is exactly what we need for 1/n^18. So, let's carefully break this down.

Again, we'll be using the power of a power rule. This rule is super versatile and shows up often when dealing with exponents. We have n raised to the power of -9, and then the whole thing is raised to the power of -2. This means we need to multiply the exponents -9 and -2.

Multiplying these exponents gives us:

(n-9)-2 = n^((-9)*(-2)) = n^18

Hey, wait a second! This looks familiar. Just like option A, option B simplifies to n^18. We did the math right, but it didn't give us the result we were hoping for. The two negative signs canceled each other out, resulting in a positive exponent. It's like when you take two steps forward and then two steps backward – you end up in the same place. So, option B is another near miss. We're learning a lot about how exponents work, but we still haven't found our winner. Don't worry, we've got more options to explore!

Unraveling Option C: (n-6)-3

Let's move on to option C: (n-6)-3. We're still in the hunt for an expression that simplifies to 1/n^18, and this one looks promising with those negative exponents. We know that negative exponents can help us get to that fraction form, so let's see what happens when we simplify this expression.

Just like the previous options, we'll use the power of a power rule. This rule is our trusty companion in this exponent adventure. We have n raised to the power of -6, and then the entire term is raised to the power of -3. So, we need to multiply these exponents together.

Multiplying the exponents -6 and -3, we get:

(n-6)-3 = n^((-6)*(-3)) = n^18

Well, well, well... it seems we have another case of the exponents canceling each other out! Option C also simplifies to n^18. We're starting to see a pattern here. When we multiply two negative exponents, we end up with a positive exponent. This is good to keep in mind as we move forward. So, option C is not the expression we're looking for. It's like we're following a treasure map, but it keeps leading us to the wrong spot. Let's hope our final option leads us to the hidden treasure!

Discovering the Solution: Option D (n-3)6

Finally, we arrive at option D: (n-3)6. This is our last chance to find the expression that simplifies to 1/n^18. We've learned a lot from the previous options, and now we can apply that knowledge to this one. Let's break it down and see if it's the answer we've been searching for.

As with the other options, we'll use the power of a power rule. We're getting really good at using this rule, aren't we? We have n raised to the power of -3, and then that whole term is raised to the power of 6. So, we need to multiply the exponents -3 and 6.

Multiplying the exponents gives us:

(n-3)6 = n^((-3)*6) = n^-18

Okay, this is interesting! We've got n^-18. But remember, we're looking for 1/n^18. How do we get there? Well, this is where our knowledge of negative exponents comes in handy. A negative exponent means we take the reciprocal of the base raised to the positive exponent. In other words:

n^-18 = 1/n^18

Eureka! We've found it! Option D, (n-3)6, simplifies to 1/n^18. It's like we've finally reached the end of our treasure map and discovered the hidden gold. This option had the perfect combination of exponents to get us to our desired result.

Final Answer

After carefully analyzing all the options, we've successfully identified the expression that simplifies to 1/n^18. The correct answer is:

D. (n-3)6

This problem was a fantastic journey through the world of exponents. We refreshed our understanding of the power of a power rule, the magic of negative exponents, and how to apply these rules to simplify complex expressions. Math can be like a puzzle, and each option is a piece. By carefully examining each piece, we can put the puzzle together and find the solution. Great job, everyone, for sticking with it and solving this problem with me!