Simplify And Evaluate (9t^3 + 4t^3 - 3t^3) / (11t - 5t) At T=6

#H1 Introduction

In mathematics, simplifying algebraic expressions is a fundamental skill. It allows us to manipulate complex expressions into more manageable forms, making them easier to understand and work with. Once simplified, we can then evaluate these expressions by substituting specific values for the variables. This article focuses on the process of simplifying the algebraic expression and then finding its value when t=6. We will break down each step, providing a clear and detailed explanation to ensure you grasp the underlying concepts. Understanding these processes is crucial for success in algebra and beyond. Therefore, mastering the ability to simplify and evaluate expressions is essential for further studies in mathematics and related fields. This article aims to provide a comprehensive guide, ensuring you can confidently tackle similar problems in the future. By following the steps and explanations provided, you'll gain a solid foundation in algebraic manipulation and evaluation.

H2 Simplifying the Algebraic Expression

H3 Combining Like Terms in the Numerator

The first step in simplifying the given expression involves combining like terms in the numerator. Like terms are terms that have the same variable raised to the same power. In the numerator, we have 9t^3, 4t^3, and -3t^3. All these terms have the variable 't' raised to the power of 3, making them like terms. To combine them, we simply add or subtract their coefficients. The coefficients are the numerical parts of the terms (9, 4, and -3 in this case). So, we perform the operation: 9 + 4 - 3. This simplifies to 10. Therefore, combining the like terms in the numerator results in 10t^3. This step is crucial because it reduces the complexity of the expression, making subsequent steps easier. By combining like terms, we consolidate the expression into a more concise form, which is essential for simplification. This simplification not only makes the expression easier to read but also reduces the chances of errors in further calculations. Understanding how to combine like terms is a foundational skill in algebra, and mastering it will greatly aid in solving more complex problems. This initial step sets the stage for the rest of the simplification process, so it’s important to perform it accurately. The result, 10t^3, will be used in the next steps to further simplify the expression.

H3 Simplifying the Denominator

The next step in simplifying the expression involves simplifying the denominator. The denominator in our expression is 11t - 5t. These are also like terms because they both have the same variable, 't', raised to the same power (which is 1 in this case). To simplify, we subtract the coefficients: 11 - 5. This results in 6. Therefore, the simplified denominator is 6t. Simplifying the denominator is just as important as simplifying the numerator. By doing so, we make the expression as a whole easier to work with. This step ensures that we are dealing with the most basic form of the denominator, which is crucial for the next stage of simplification, where we will look at canceling out common factors. A simplified denominator not only makes the overall expression cleaner but also reduces the potential for errors in later calculations. The ability to simplify expressions by combining like terms, whether in the numerator or the denominator, is a fundamental skill in algebra. It allows us to manipulate equations and expressions into more manageable forms, which is essential for solving problems and understanding mathematical concepts. The simplified form of the denominator, 6t, is now ready to be used in the final simplification of the entire algebraic expression.

H3 Reducing the Fraction

After simplifying both the numerator and the denominator, we now have the expression 10t^3 / 6t. The final step in simplifying the expression is to reduce the fraction by canceling out common factors. To do this, we look for the greatest common divisor (GCD) of the coefficients (10 and 6) and any common variables. The GCD of 10 and 6 is 2, so we can divide both the numerator and the denominator by 2. This gives us 5t^3 / 3t. Next, we look at the variables. Both the numerator and the denominator have 't'. In the numerator, we have t^3 (which means t * t * t), and in the denominator, we have t (which means t^1). We can cancel out one 't' from both the numerator and the denominator. This leaves us with t^2 in the numerator. So, after canceling out the common factors, the simplified expression becomes 5t^2 / 3. Reducing fractions is a critical skill in algebra because it allows us to express the algebraic expression in its simplest form. This not only makes the expression easier to understand but also makes it easier to work with in further calculations or problem-solving scenarios. By identifying and canceling common factors, we ensure that we are dealing with the most concise representation of the expression. The process of reducing fractions involves both numerical and algebraic simplification, and it is a fundamental technique that is used extensively in mathematics. The final simplified form, 5t^2 / 3, is now ready for evaluation, which is the next step in our process.

H2 Evaluating the Simplified Expression

H3 Substituting the Value of t

Now that we have simplified the expression to 5t^2 / 3, the next step is to evaluate it when t = 6. This involves substituting the value of 't' into the simplified expression. We replace every instance of 't' with the number 6. So, the expression becomes 5 * (6^2) / 3. Substitution is a fundamental operation in algebra, allowing us to find the numerical value of an expression for a given value of the variable. It’s crucial to perform the substitution accurately to ensure the correct final result. When substituting, it is often helpful to use parentheses to clearly indicate the substitution, especially when dealing with more complex expressions. This helps to avoid confusion and potential errors in calculation. The act of substituting the value of the variable transforms the algebraic expression into a numerical expression, which we can then evaluate using arithmetic operations. This process is essential for solving equations, modeling real-world scenarios, and making predictions based on mathematical relationships. Substituting the value of t is a straightforward process, but it is a key step in determining the final value of the expression. Once the substitution is complete, we proceed to simplify the numerical expression, following the order of operations to ensure an accurate result.

H3 Calculating the Result

After substituting t = 6 into the simplified expression, we have 5 * (6^2) / 3. To calculate the result, we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). First, we evaluate the exponent: 6^2 means 6 multiplied by itself, which equals 36. So the expression becomes 5 * 36 / 3. Next, we perform the multiplication: 5 * 36 equals 180. Now the expression is 180 / 3. Finally, we perform the division: 180 divided by 3 equals 60. Therefore, the value of the expression when t = 6 is 60. Calculating the result accurately requires a good understanding of the order of operations. Following PEMDAS ensures that we perform the operations in the correct sequence, leading to the correct answer. In this case, evaluating the exponent first, then multiplying, and finally dividing gives us the accurate result. Arithmetic skills are essential in algebra, and being able to perform these calculations efficiently is key to solving algebraic problems. The process of evaluating an expression after substitution provides a numerical answer that represents the value of the expression under specific conditions. This result can be used for various purposes, such as verifying solutions to equations or making predictions in mathematical models. The final answer, 60, represents the value of the original expression when t is equal to 6.

H1 Conclusion

In this article, we successfully simplified the algebraic expression (9t^3 + 4t^3 - 3t^3) / (11t - 5t) and found its value when t = 6. The process involved several key steps, including combining like terms in both the numerator and the denominator, reducing the fraction by canceling out common factors, substituting the value of t into the simplified expression, and finally, calculating the result using the order of operations. Each step is crucial in ensuring the correct final answer. Simplifying algebraic expressions is a fundamental skill in mathematics. It allows us to transform complex expressions into simpler, more manageable forms. This not only makes the expressions easier to understand but also reduces the chances of errors in calculations. The ability to combine like terms, reduce fractions, and follow the order of operations are all essential components of algebraic manipulation. Evaluating expressions by substituting values for variables is another critical skill. It enables us to find specific values of expressions under given conditions, which is essential for solving equations and applying mathematical concepts to real-world problems. By mastering these skills, you will be well-equipped to tackle more complex algebraic problems and excel in your mathematical studies. The final answer, 60, represents the numerical value of the original expression when t is equal to 6, demonstrating the practical application of algebraic simplification and evaluation. This comprehensive guide has provided a step-by-step approach to solving this type of problem, equipping you with the knowledge and skills needed to succeed in algebra.