Simplifying Expressions: Math Problem Solved!

Hey everyone! Today, we're diving into a fun math problem: simplifying expressions. It might sound a bit intimidating at first, but trust me, it's like solving a puzzle. We'll break down the question step-by-step, making sure you grasp every concept. So, grab your pencils and let's get started. We'll tackle this expression: What expressions equal the product of $10x^9$ and $\left(60x^{-6}\right)^{-1}$? And don't worry, we'll keep it simple and easy to follow. Remember, understanding these basics is crucial for all kinds of math, so let's make sure we get it right, shall we?

Deciphering the Core Problem

Alright, guys, let's break down the main problem: We need to find expressions equal to the product of $10x^9$ and $\left(60x^{-6}\right)^{-1}$. To make this easier, let's first simplify the expression $\left(60x^{-6}\right)^{-1}$. Remember that a negative exponent means we take the reciprocal. Therefore, $\left(60x^{-6}\right)^{-1}$ is the same as $\frac{1}{60x^{-6}}$. Now, we can rewrite the original problem as finding expressions equivalent to $10x^9 \cdot \frac{1}{60x^{-6}}$. We're essentially multiplying the first term, $10x^9$, by the reciprocal of $60x^{-6}$. It's all about playing with those exponents and understanding what they mean. The key here is to simplify. We'll simplify this expression by combining like terms and applying the rules of exponents, like how to deal with negative exponents. Always try to simplify step by step, which will help you avoid making mistakes and will help you get the correct answer faster. Pay close attention to how the exponents change and how the constants combine. Let's make sure we do it right!

Step-by-Step Simplification

Okay, let's get into the nitty-gritty and simplify this expression. First off, let's address the $\frac{1}{60x^{-6}}$ part. Remember that $x^{-6}$ is the same as $\frac{1}{x^6}$. So, we can rewrite the expression as $\frac{1}{60} \cdot x^6$ (since the negative exponent moves the $x$ to the numerator, and the $60$ stays in the denominator). This simplifies to $\frac{x^6}{60}$. Now, our original problem becomes $10x^9 \cdot \frac{x^6}{60}$. Next, multiply the terms. Multiply the coefficients (the numbers). We have $10$ in the numerator and $60$ in the denominator, so we can simplify the fraction. $10$ divided by $60$ equals $\frac{1}{6}$. Then, multiply the variables. When multiplying exponents with the same base, you add the powers. So, $x^9 \cdot x^6$ becomes $x^{9+6}$, which equals $x^{15}$. Combining all these, we get $\frac{1}{6}x^{15}$ or $\frac{x^{15}}{6}$. We're well on our way to identifying the correct answer. We have effectively simplified the given expression. Always remember to simplify the fraction first if possible. Always try to cancel and reduce the numbers to their simplest form. That would make the calculation a lot easier. Now, we are ready to compare our simplified expression with the given options.

Evaluating the Answer Choices

Now, let's look at the answer choices. We need to find the expressions that are equal to $\frac{x^{15}}{6}$.

• A. $\frac{10x^9}{60x^{-6}}$: Let's simplify this. The $10$ over $60$ simplifies to $\frac{1}{6}$. The $x^{-6}$ in the denominator becomes $x^6$ when brought to the numerator. So, we get $\frac{1}{6}x^9 \cdot x^6$, which equals $\frac{x^{15}}{6}$. This one's a keeper!
• B. $\frac{10x^9}{60x^6}$: Here, we have $x^6$ in the denominator. So, when simplifying, we get $\frac{1}{6}x^{9-6}$, which simplifies to $\frac{1}{6}x^3$ or $\frac{x^3}{6}$. Nope, not our answer.
• C. $\frac{x^{15}}{6}$: This matches our simplified expression exactly. Bingo!
• D. $6x^{15}$: This is not equal to $\frac{x^{15}}{6}$. No, this one is not correct. We are looking for the expressions that match our simplified form, and this does not.

So, after a thorough evaluation, we can say that A and C are the correct answers. We've simplified, evaluated, and now we know the correct options. That wasn't so bad, right?

Key Takeaways and Tips for Success

Here are some of the main things to remember. Always Simplify: Simplifying expressions is all about breaking them down step by step. Understand Exponents: Negative exponents, adding exponents when multiplying, and subtracting when dividing – get these rules down. Be Organized: Write your work clearly, and don't skip steps. This helps avoid mistakes. Practice: The more problems you solve, the better you get. Try different problems to hone your skills. Check Your Work: Always double-check your answers. Going back and re-evaluating each step helps catch any potential errors. Keep practicing and applying these tips. It is important to solve problems step by step and understand the concepts to get good at mathematics.

Final Thoughts

Great job, everyone! You've successfully navigated a math problem involving expressions and exponents. Remember, practice is key, and each problem you solve builds your confidence. Keep exploring, keep learning, and don’t be afraid to tackle new challenges. You've got this, and with consistent effort, you'll become more and more comfortable with these types of problems. Thanks for joining me today, and keep up the fantastic work!