Evaluate 12(2)^(x-2) For X=6.5 To The Nearest Thousandth

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In the realm of mathematics, exponential expressions play a crucial role in modeling various phenomena, from population growth to radioactive decay. Understanding how to evaluate these expressions is fundamental to grasping their implications and applications. This comprehensive guide will walk you through the process of evaluating the exponential expression $12(2)^{x-2}$ for $x=6.5$, providing a clear and concise approach that can be applied to a wide range of similar problems.

Understanding Exponential Expressions

Before we delve into the evaluation process, let's take a moment to understand the anatomy of an exponential expression. An exponential expression consists of a base raised to a power, often referred to as the exponent. The base represents the quantity that is being multiplied repeatedly, while the exponent indicates the number of times the base is multiplied by itself. In the expression $12(2)^{x-2}$, the base is 2, and the exponent is $x-2$. The coefficient 12 is a constant that multiplies the entire exponential term.

Exponential expressions are governed by a set of rules and properties that dictate how they behave under various operations. One of the most important rules is the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This rule dictates the sequence in which operations should be performed to arrive at the correct result. In the case of exponential expressions, we must first evaluate the exponent before performing any multiplication or addition.

Step 1: Substitute the Value of x

The first step in evaluating the expression $12(2)^{x-2}$ for $x=6.5$ is to substitute the given value of x into the expression. This means replacing the variable x with the numerical value 6.5. After the substitution, the expression becomes $12(2)^{6.5-2}$. This step is crucial as it transforms the expression from a general form to a specific numerical calculation.

Substitution is a fundamental concept in algebra and calculus, allowing us to evaluate expressions and functions for specific values of their variables. It's like plugging in a particular input into a machine to see what output it produces. In this case, we are plugging in the value 6.5 for x to determine the value of the exponential expression at that specific point.

Step 2: Simplify the Exponent

Now that we have substituted the value of x, our expression is $12(2)^{6.5-2}$. The next step is to simplify the exponent, which is $6.5-2$. Performing this subtraction, we get $4.5$. So, the expression now becomes $12(2)^{4.5}$. This step is essential because it reduces the complexity of the expression and makes it easier to evaluate. By simplifying the exponent, we are essentially determining the power to which the base 2 is raised.

The order of operations (PEMDAS) dictates that we must simplify the exponent before performing any multiplication. This is because the exponentiation operation has a higher precedence than multiplication. By simplifying the exponent first, we ensure that we are calculating the correct power of 2 before multiplying it by 12.

Step 3: Evaluate the Exponential Term

The heart of evaluating the exponential expression lies in calculating the value of the exponential term, which is $2^{4.5}$. This means raising the base 2 to the power of 4.5. Since the exponent is not an integer, we'll need to use a calculator to find the value of $2^{4.5}$. Using a calculator, we find that $2^{4.5} ext{ ≈ } 22.627417$.

This step showcases the power of exponents in representing repeated multiplication. The exponent 4.5 indicates that we are multiplying 2 by itself 4.5 times. While it's impossible to multiply a number by itself a fractional number of times in the traditional sense, the concept of fractional exponents is well-defined in mathematics and can be calculated using calculators or mathematical software. The result, approximately 22.627417, is the value of the exponential term $2^{4.5}$.

Step 4: Multiply by the Coefficient

With the exponential term $2^{4.5}$ evaluated to approximately 22.627417, our expression now looks like $12 imes 22.627417$. The final step is to multiply this value by the coefficient 12. Performing this multiplication, we get $12 imes 22.627417 ext{ ≈ } 271.529004$. This step combines the exponential term with the constant factor, giving us the final value of the expression.

Multiplication is the last operation we perform in this evaluation, following the order of operations (PEMDAS). By multiplying the exponential term by the coefficient, we are scaling the result of the exponential calculation. The coefficient acts as a multiplier, increasing or decreasing the value of the exponential term depending on its magnitude.

Step 5: Round to the Nearest Thousandth

The question asks us to express the answer to the nearest thousandth. Our current result is approximately 271.529004. To round this to the nearest thousandth, we look at the digit in the ten-thousandths place, which is 0. Since 0 is less than 5, we round down, keeping the digit in the thousandths place as it is. Therefore, the final answer rounded to the nearest thousandth is 271.529.

Rounding is a crucial skill in mathematics and scientific calculations, allowing us to express numerical results with appropriate precision. Rounding to the nearest thousandth means expressing the result with three decimal places, which is often sufficient for practical applications. By rounding our answer, we are simplifying the result while maintaining a reasonable level of accuracy.

Final Answer

Therefore, the value of the expression $12(2)^{x-2}$ for $x=6.5$, expressed to the nearest thousandth, is 271.529.

This step-by-step guide has demonstrated the process of evaluating an exponential expression, highlighting the importance of the order of operations and the use of calculators for non-integer exponents. By following these steps, you can confidently evaluate a wide range of exponential expressions and apply them to various mathematical and scientific contexts.

Key Takeaways

  • Order of Operations (PEMDAS): Remember to follow the order of operations (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) when evaluating expressions.
  • Substitution: Replace variables with their given values to transform expressions into numerical calculations.
  • Simplifying Exponents: Simplify exponents before performing other operations to reduce the complexity of the expression.
  • Calculator Usage: Use calculators to evaluate exponential terms with non-integer exponents.
  • Rounding: Round your final answer to the specified level of precision.

By mastering these key takeaways, you will be well-equipped to tackle various exponential expression evaluations and unlock their potential in mathematical problem-solving.

Practice Problems

To solidify your understanding of evaluating exponential expressions, try working through these practice problems:

  1. Evaluate $5(3)^{x+1}$ for $x=2.8$, rounded to the nearest hundredth.
  2. Evaluate $10(0.5)^{x-3}$ for $x=7.2$, rounded to the nearest tenth.
  3. Evaluate $8(4)^{x/2}$ for $x=5.5$, rounded to the nearest thousandth.

By practicing these problems, you will gain confidence in your ability to evaluate exponential expressions and apply them to different scenarios.

Conclusion

Evaluating exponential expressions is a fundamental skill in mathematics with applications across various fields. By understanding the order of operations, using calculators when necessary, and practicing regularly, you can master this skill and unlock the power of exponential functions. This guide has provided a comprehensive step-by-step approach to evaluating exponential expressions, empowering you to confidently tackle a wide range of problems.