Unlocking Rectangle & Algebraic Secrets: A Math Adventure!

Hey math enthusiasts! Let's dive into some cool problems today. We've got a geometry puzzle about rectangles and an algebra challenge involving exponents. Get ready to flex those brain muscles! We'll break down each problem step-by-step, making sure you grasp the concepts. So, grab your pencils and let's get started. This is gonna be fun, guys!

Solving the Rectangle Riddle: Perimeter Power!

Let's tackle the rectangle problem first. The key information here is that the ratio of the length to the width of a rectangle is 5:3, and its area is 240 cm². Our goal? To find the perimeter. Sounds like a piece of cake, right? Well, let's see how we can solve it. Remember that understanding the fundamental concepts is key to problem-solving in mathematics. We are going to utilize the concept of ratio and proportion to solve this problem. Understanding of the area and perimeter of a rectangle is also another important aspect of solving the problem. So let's get into it.

First, let's represent the length and width using the ratio. Since the ratio is 5:3, we can say the length is 5x and the width is 3x, where 'x' is a constant. Remember that the area of a rectangle is calculated by multiplying its length and width. We know that the area is 240 cm², so we can set up the equation: (5x) * (3x) = 240. Now, let's simplify and solve for 'x'. Multiplying 5x and 3x gives us 15x². So, our equation becomes 15x² = 240. To find x², we divide both sides by 15: x² = 240 / 15, which simplifies to x² = 16. Finally, to find x, we take the square root of both sides: x = √16, meaning x = 4.

Now that we know x = 4, we can find the actual length and width of the rectangle. The length is 5x, so it's 5 * 4 = 20 cm. The width is 3x, so it's 3 * 4 = 12 cm. Great! We've got the dimensions. The last step is to calculate the perimeter. The perimeter of a rectangle is found using the formula: P = 2 * (length + width). Plugging in our values, we get P = 2 * (20 cm + 12 cm). Simplifying, we have P = 2 * 32 cm, which gives us P = 64 cm. Voila! The perimeter of the rectangle is 64 cm. That wasn't so tough, right? We just needed to understand the relationship between the ratio, area, and dimensions of a rectangle, and we were golden. Always remember to check your work and make sure your answer makes sense in the context of the problem. In this case, 64 cm for the perimeter seems reasonable given the area of 240 cm².

So, the correct answer is d. 64 cm. This kind of problem is common in geometry and tests your ability to apply formulas and work with ratios and equations. Keep practicing, and you'll become a pro at these problems! We can understand that mathematical problems are not just about finding the correct answer, but also about the process of learning and understanding mathematical concepts. Now, let's move on to the next problem, we're gonna learn something interesting.

Decoding Exponents: An Algebraic Adventure!

Alright, let's switch gears and tackle an algebra problem. We're asked to simplify the expression $(4p^6)^2 : 2p^3$. This involves working with exponents and powers. Don't worry if it looks intimidating at first; we'll break it down step-by-step. Remember, the key to solving these types of problems is to apply the rules of exponents correctly. We'll start with the first part of the expression, $(4p^6)^2$. The power of a product rule says that when you have a term inside parentheses raised to a power, you apply that power to each factor inside. In this case, we have to apply the power of 2 to both 4 and p^6.

So, $(4p^6)^2$ becomes 4² * (p^6)². Now, calculate 4², which is 4 * 4 = 16. And for (p^6)², we use the power of a power rule, which states that when you have a power raised to another power, you multiply the exponents. So, (p^6)² becomes p^(6*2) = p¹². Therefore, $(4p^6)^2$ simplifies to 16p¹². Now, our original expression has become 16p¹² : 2p³. The division part of the problem comes next. To divide terms with exponents, you divide the coefficients and subtract the exponents of the variables. In other words, divide 16 by 2, which gives us 8. Then, subtract the exponent of p in the denominator (3) from the exponent of p in the numerator (12). So, p¹² : p³ becomes p^(12-3) = p^9. Putting it all together, 16p¹² : 2p³ simplifies to 8p^9. Easy peasy, right?

So, the correct answer is not listed. However, based on our calculations, the correct answer should be 8p^9. This problem tests your knowledge of exponent rules, including the power of a product, power of a power, and division of exponents. Make sure you're comfortable with these rules, as they are fundamental in algebra. Remember to practice these rules regularly, and you'll find that simplifying expressions like this becomes second nature. Keep in mind that understanding and applying the rules of exponents is fundamental to working with algebraic expressions and equations. This will help you to become confident in solving problems, and make the subject less intimidating.

Key Takeaways: Mastering Math Problems!

• Ratios and Rectangles: Always relate ratios to actual dimensions and use the area formula to find missing sides. The perimeter is easily calculated once you know the length and width.
• Exponents are Your Friends: Understand the power of a product, the power of a power, and the division rules for exponents to simplify expressions confidently. Practice makes perfect in exponent problems!
• Step-by-Step is the Way: Break down complex problems into smaller, manageable steps. This makes the solution process easier and reduces the chances of errors.
• Check Your Answers: Always review your work and make sure your answers are reasonable and make sense in the context of the problem. This helps prevent silly mistakes.
• Practice, Practice, Practice: The more you practice, the more confident you'll become in solving math problems. Work through various examples and try different types of problems to strengthen your skills. Get used to the pattern of these types of questions.

That's all for today, guys! I hope you had fun solving these problems with me. Keep practicing, and remember, math can be enjoyable. See you next time for more math adventures! Keep your math skills sharp, and don't be afraid to ask for help if you need it. Remember, mathematics is about exploring concepts and expanding your problem-solving abilities. Have a great day!