Evaluating The Expression 10 + 16r² For R = 7/2
In this article, we will walk through the process of evaluating the algebraic expression 10 + 16r² for a given value of r. Specifically, we will substitute r = 7/2 into the expression and simplify it step-by-step to arrive at the final numerical result. This exercise is fundamental in algebra and helps to reinforce the understanding of variable substitution and order of operations.
Understanding the Expression
The expression 10 + 16r² is a polynomial expression. It consists of two terms: a constant term (10) and a variable term (16r²). The variable term includes the variable r raised to the power of 2 (which means r squared) and multiplied by the coefficient 16. To evaluate this expression for a specific value of r, we need to replace r with that value and perform the arithmetic operations following the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
Before we dive into the calculations, let’s break down each component of the expression. The constant term 10 remains unchanged regardless of the value of r. The term 16r² is where the substitution will take place. The exponent ()2 indicates that we need to square the value of r before multiplying it by 16. This order is crucial to obtain the correct result. The coefficient 16 scales the squared value of r, thereby influencing the overall value of the expression.
Understanding these individual components allows us to approach the evaluation in a systematic manner. First, we substitute the given value of r into the expression. Second, we simplify the exponent. Third, we perform the multiplication. Finally, we complete the calculation by adding the constant term. This methodical approach helps to minimize errors and ensures accuracy in our evaluation. Now, let’s proceed with substituting r = 7/2 into the expression 10 + 16r².
Substituting r = 7/2
The next crucial step in evaluating the expression 10 + 16r² is substituting the given value of r, which is 7/2, into the expression. This substitution is a fundamental operation in algebra, where we replace a variable with its numerical value to transform an algebraic expression into an arithmetic expression that can be simplified to a single numerical value. In our case, we will replace every instance of r in the expression with the fraction 7/2.
So, starting with the original expression 10 + 16r², we replace r with 7/2, which gives us 10 + 16(7/2)². It is essential to enclose the substituted value 7/2 in parentheses to ensure that the exponent (2) applies only to the fraction 7/2 and not just to the numerator 7. This correct placement of parentheses is vital for maintaining the integrity of the expression and following the order of operations accurately.
Now, the expression looks like an arithmetic problem that can be solved by following the standard rules of arithmetic. We have a constant (10), a multiplication (16 times the square of 7/2), and an addition operation. Remember, the order of operations (PEMDAS/BODMAS) dictates that we must address the exponent first before performing the multiplication and addition. This principle is crucial to ensure that we arrive at the correct final answer. The next step involves squaring the fraction 7/2, which will simplify the expression further and bring us closer to the solution. Let’s proceed with squaring 7/2 in the next section.
Squaring 7/2
Before we can complete the evaluation of the expression 10 + 16(7/2)², we need to address the exponent. Specifically, we must calculate the square of the fraction 7/2. Squaring a fraction means multiplying the fraction by itself. In mathematical terms, (7/2)² is equivalent to (7/2) * (7/2).
When multiplying fractions, we multiply the numerators together to get the new numerator, and we multiply the denominators together to get the new denominator. So, (7/2) * (7/2) = (7 * 7) / (2 * 2). Calculating the products, we have 7 * 7 = 49 and 2 * 2 = 4. Therefore, (7/2)² simplifies to 49/4. This step is crucial because it transforms the exponential term into a simple fraction, making it easier to proceed with the rest of the calculation.
Now that we have determined that (7/2)² = 49/4, we can substitute this value back into our expression. Our expression now looks like this: 10 + 16(49/4). The next operation according to the order of operations is multiplication. We need to multiply 16 by the fraction 49/4. This multiplication is the next step in simplifying the expression and will bring us closer to the final answer. Let’s proceed with the multiplication in the next section.
Multiplying 16 by 49/4
Having simplified the exponent in the expression 10 + 16(49/4), our next step is to perform the multiplication. We need to multiply the whole number 16 by the fraction 49/4. When multiplying a whole number by a fraction, it can be helpful to think of the whole number as a fraction with a denominator of 1. So, we can rewrite 16 as 16/1.
Now, we have the multiplication (16/1) * (49/4). To multiply fractions, we multiply the numerators together and the denominators together. This gives us (16 * 49) / (1 * 4). Before we perform the full multiplication, we can simplify the expression to make the calculation easier. Notice that 16 and 4 have a common factor of 4. We can divide both 16 and 4 by 4 to simplify. 16 divided by 4 is 4, and 4 divided by 4 is 1. So, our expression simplifies to (4/1) * (49/1).
Now the multiplication is much simpler: (4 * 49) / (1 * 1). Multiplying the numerators, we have 4 * 49 = 196. Multiplying the denominators, we have 1 * 1 = 1. Thus, (4 * 49) / (1 * 1) = 196/1, which is simply 196. This multiplication step has significantly simplified our expression. Now we know that 16(49/4) equals 196. Substituting this back into our expression, we now have 10 + 196. The final step is to perform the addition.
Adding 10 and 196
We have now simplified the expression 10 + 16(7/2)² to the point where we just need to perform the addition. Our expression is now 10 + 196. This is a straightforward addition of two whole numbers.
Adding 10 and 196 is a basic arithmetic operation. 10 plus 196 equals 206. Therefore, the final result of evaluating the expression 10 + 16(7/2)² when r = 7/2 is 206. This completes the evaluation process, and we have arrived at the numerical value of the expression for the given value of r.
Final Result
After carefully following the steps of substitution and simplification, we have arrived at the final result. We started with the expression 10 + 16r² and substituted r = 7/2. We then squared 7/2 to get 49/4, multiplied 16 by 49/4 to get 196, and finally, added 10 to 196. The result of this entire process is 206.
Therefore, the value of the expression 10 + 16r² when r = 7/2 is 206. This entire process demonstrates the importance of understanding and correctly applying the order of operations (PEMDAS/BODMAS) in algebra. Each step, from substitution to simplification, is crucial in arriving at the accurate final answer. By breaking down the problem into manageable steps, we have successfully evaluated the expression.
