Solving Math Problems Vicky And Rakesh's Scores And Expression Simplification

In this article, we delve into two intriguing mathematical problems. The first involves determining the marks scored by Vicky and Rakesh in Maths, given a relationship between their scores and the product of their marks. The second problem focuses on simplifying a complex algebraic expression involving variables with exponents. Both problems require a solid understanding of basic mathematical principles and algebraic manipulation. Let's embark on this mathematical journey to unravel the solutions step by step.

Problem 1: Unraveling Vicky and Rakesh's Math Scores

The first challenge we face is a fascinating problem involving Vicky and Rakesh's scores in mathematics. We are told that Vicky scored x marks, while Rakesh, with his mathematical prowess, secured x³ marks. The problem adds an intriguing twist by stating that the product of their marks is 256. This sets up an equation that we can solve to find the value of x, and subsequently, the individual scores of Vicky and Rakesh. To solve this, we will translate the word problem into a mathematical equation. The product of their marks can be represented as x * x³ = 256. This equation forms the foundation for our solution.

Setting up the Equation

In this section, we will explore the equation in detail. The equation x * x³ = 256 represents the core relationship between Vicky and Rakesh's scores. To solve this equation, we need to simplify it using the laws of exponents. Recall that when multiplying terms with the same base, we add their exponents. In this case, we have x¹ * x³ which simplifies to x¹⁺³ = x⁴. Now, our equation becomes x⁴ = 256. This simplified equation is much easier to handle and allows us to move closer to finding the value of x. The next step involves finding the fourth root of 256.

Solving for x

To determine the value of x, we need to find the fourth root of 256. In other words, we are looking for a number that, when raised to the power of 4, equals 256. We can express this mathematically as x = ⁴√256. To find this root, we can consider the prime factorization of 256. The prime factorization of 256 is 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2, which can be written as 2⁸. Now, we can rewrite the fourth root of 256 as ⁴√2⁸. Using the properties of exponents and roots, we can simplify this further. Recall that ⁴√2⁸ is equivalent to 2^(8/4), which equals 2². Therefore, x = 2² = 4. We have successfully found the value of x, which is 4. This value is crucial as it allows us to determine the individual scores of Vicky and Rakesh.

Determining Vicky and Rakesh's Scores

Now that we have found the value of x, we can easily calculate the marks scored by Vicky and Rakesh. Vicky's score is given by x, which we now know is 4. Therefore, Vicky scored 4 marks in Maths. Rakesh's score is given by x³, which means we need to calculate 4³. This is 4 * 4 * 4, which equals 64. So, Rakesh scored 64 marks in Maths. It's interesting to note the significant difference in their scores, with Rakesh scoring considerably higher than Vicky. This problem demonstrates how algebraic equations can be used to represent and solve real-world scenarios, in this case, the scores of two students in a subject.

Problem 2: Simplifying a Complex Algebraic Expression

The second problem presents a challenge in simplifying a complex algebraic expression. The expression is 3x⁻²y⁻³z² × 5x²x y z³ + x³y²z⁻¹. This expression involves variables with both positive and negative exponents, as well as multiplication and addition operations. To simplify this expression effectively, we will need to apply the laws of exponents and the distributive property. The goal is to combine like terms and reduce the expression to its simplest form. This type of problem is common in algebra and requires careful attention to detail and a strong understanding of algebraic principles.

Breaking Down the Expression

Let's dissect the given expression to understand its components. The expression is 3x⁻²y⁻³z² × 5x²x y z³ + x³y²z⁻¹. We can identify two main parts: the product of the first two terms and the addition of the third term. The first part, 3x⁻²y⁻³z² × 5x²x y z³, involves the multiplication of several terms with exponents. The second part, x³y²z⁻¹, is a single term with variables and exponents. To simplify the entire expression, we will first focus on simplifying the product in the first part. This involves multiplying the coefficients and applying the laws of exponents to the variables.

Simplifying the Product

Now, let's simplify the product 3x⁻²y⁻³z² × 5x²x y z³. To do this, we will multiply the coefficients (the numerical parts) and then multiply the variables with the same base by adding their exponents. First, multiply the coefficients: 3 × 5 = 15. Next, let's consider the x terms: x⁻² * x² * x¹ = x⁻²⁺²⁺¹ = x¹. Now, let's look at the y terms: y⁻³ * y¹ = y⁻³⁺¹ = y⁻². Finally, let's consider the z terms: z² * z³ = z²⁺³ = z⁵. Combining these results, the product simplifies to 15x¹y⁻²z⁵. This simplified term will now be added to the third term in the original expression.

Combining Terms

Now that we have simplified the product, we can combine it with the third term in the original expression. The expression now looks like this: 15x¹y⁻²z⁵ + x³y²z⁻¹. To complete the simplification, we need to determine if there are any like terms that can be combined. Like terms are terms that have the same variables raised to the same powers. In this case, we have the term 15x¹y⁻²z⁵ and the term x³y²z⁻¹. Comparing these terms, we can see that they do not have the same variables raised to the same powers. The exponents of x, y, and z are different in each term. Therefore, these terms cannot be combined further. The expression 15x¹y⁻²z⁵ + x³y²z⁻¹ is the simplest form of the given expression.

In this article, we successfully tackled two distinct mathematical problems. The first problem required us to find the marks scored by Vicky and Rakesh, using an equation derived from the product of their scores. We found that Vicky scored 4 marks and Rakesh scored 64 marks. The second problem challenged us to simplify a complex algebraic expression involving variables with exponents. By applying the laws of exponents and simplifying the expression step by step, we arrived at the simplest form: 15x¹y⁻²z⁵ + x³y²z⁻¹. These problems highlight the importance of understanding fundamental mathematical concepts and applying them methodically to solve complex problems. Whether it's unraveling the scores of students or simplifying algebraic expressions, mathematics provides a powerful framework for problem-solving.