Simplifying (-11r¹⁰h)(5rh³) An Algebraic Expression
Introduction
In this comprehensive article, we will delve into the process of simplifying the algebraic expression (-11r¹⁰h)(5rh³). This problem falls under the category of mathematics, specifically algebra, and involves the application of fundamental rules of exponents and multiplication. Our primary goal is to break down the expression step by step, ensuring a clear and understandable explanation for readers of all backgrounds. By the end of this article, you will have a solid grasp of how to simplify such expressions, a skill that is crucial in various mathematical and scientific contexts. We'll start by identifying the key components of the expression, then proceed with the multiplication, and finally, simplify the resulting terms to arrive at the most concise form. So, let's embark on this mathematical journey and unravel the intricacies of algebraic simplification.
Breaking Down the Expression
To effectively simplify the expression (-11r¹⁰h)(5rh³), we first need to identify its individual components and understand how they interact with each other. The expression consists of two terms enclosed in parentheses, which indicates that we need to multiply these terms together. The first term, -11r¹⁰h, is a product of a numerical coefficient (-11), a variable r raised to the power of 10 (r¹⁰), and another variable h. The second term, 5rh³, is similarly composed of a numerical coefficient (5), the variable r, and the variable h raised to the power of 3 (h³). Understanding this structure is crucial because it allows us to apply the rules of multiplication and exponents correctly. We'll begin by multiplying the numerical coefficients, then move on to the variables, remembering to add the exponents when multiplying like bases. This methodical approach ensures that we don't miss any steps and arrive at the correct simplified form. The key here is to treat each component separately and then combine the results in a logical manner.
Multiplying the Coefficients
The initial step in simplifying the expression (-11r¹⁰h)(5rh³) involves multiplying the numerical coefficients. The coefficients are the numerical parts of the terms, which in this case are -11 and 5. Multiplying these two numbers together is straightforward: -11 * 5 = -55. This result becomes the numerical coefficient of our simplified expression. It's crucial to pay attention to the signs of the numbers, as a negative number multiplied by a positive number yields a negative result. This single calculation forms the foundation for the rest of the simplification process. Once we have the coefficient, we can focus on the variables and their exponents, combining them according to the rules of algebra. The accurate multiplication of the coefficients is a pivotal step, as any error here will propagate through the rest of the calculation. Therefore, we ensure that this initial step is executed with precision and care.
Multiplying the Variables
After successfully multiplying the coefficients, the next crucial step in simplifying (-11r¹⁰h)(5rh³) is to multiply the variables. We have two variables in this expression: r and h. To multiply them, we need to recall the rule of exponents that states when multiplying like bases, we add their exponents. Let's start with the variable r. In the expression, we have r¹⁰ in the first term and r (which can be considered as r¹) in the second term. Multiplying these gives us r¹⁰ * r¹ = r¹⁰⁺¹ = r¹¹. Next, we consider the variable h. We have h (which is h¹) in the first term and h³ in the second term. Multiplying these gives us h¹ * h³ = h¹⁺³ = h⁴. By applying this rule, we've successfully combined the variables. Now, we can assemble the simplified expression by combining the multiplied coefficients and variables. This process highlights the importance of understanding and correctly applying the rules of exponents, which are fundamental to algebraic simplification.
Combining the Terms
Having multiplied both the coefficients and the variables, the final step in simplifying the expression (-11r¹⁰h)(5rh³) is to combine the results. We found that the product of the coefficients is -55, the product of the r terms is r¹¹, and the product of the h terms is h⁴. To combine these, we simply write them together. Therefore, the simplified expression becomes -55r¹¹h⁴. This result represents the most concise form of the original expression. By following the steps of multiplying the coefficients and then the variables, we have effectively simplified a complex expression into a much simpler one. This process demonstrates the power of algebraic manipulation and the importance of understanding the fundamental rules of exponents and multiplication. The final simplified form, -55r¹¹h⁴, clearly shows the relationship between the numerical coefficient and the variables, making it easier to interpret and use in further calculations or applications.
Final Result
In conclusion, the simplified form of the expression (-11r¹⁰h)(5rh³) is -55r¹¹h⁴. We arrived at this result by systematically breaking down the expression, multiplying the coefficients, multiplying the variables using the rules of exponents, and then combining the results. This step-by-step approach ensures accuracy and clarity in the simplification process. The final expression, -55r¹¹h⁴, is a concise and simplified representation of the original expression, making it easier to work with in various mathematical contexts. Understanding how to simplify algebraic expressions like this is a fundamental skill in mathematics, with applications in various fields such as physics, engineering, and computer science. By mastering these techniques, one can tackle more complex problems with confidence and precision. The journey from the initial expression to the simplified form underscores the elegance and power of algebraic manipulation.
