Given 2x = 64, Find The Value Of X, Then Evaluate 3x, 2x - 3, 4^(2x - 3), And (-1/3)^(-2x).

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In this comprehensive guide, we will delve into the process of solving for the variable x in the equation 2x = 64, a fundamental concept in algebra. Understanding how to isolate variables and determine their values is crucial for mastering more complex mathematical problems. Once we've successfully found the value of x, we will then apply this knowledge to evaluate a series of expressions, including linear expressions, exponential expressions, and expressions with negative exponents. This exploration will not only reinforce your algebraic skills but also demonstrate the interconnectedness of different mathematical concepts. Whether you're a student looking to solidify your understanding or someone seeking a refresher, this article will provide a step-by-step approach to solving for x and evaluating related expressions. So, let's embark on this mathematical journey and unlock the solutions together.

Solving for x in the Equation 2x = 64

In this section, we will meticulously walk through the process of solving for x in the equation 2x = 64. This equation represents a simple linear relationship, where x is multiplied by the coefficient 2. To isolate x and find its value, we need to perform the inverse operation of multiplication, which is division. The key principle here is to maintain the balance of the equation, meaning that any operation performed on one side must also be performed on the other side. This ensures that the equality remains valid throughout the solving process. Dividing both sides of the equation by the coefficient of x will effectively undo the multiplication and reveal the value of x. Let's break down the steps in detail:

  1. Start with the given equation: The initial equation we are presented with is 2x = 64. This equation states that two times the value of x is equal to 64. Our goal is to isolate x on one side of the equation to determine its numerical value.
  2. Divide both sides by the coefficient of x: The coefficient of x in this equation is 2. To isolate x, we need to divide both sides of the equation by 2. This is a crucial step in maintaining the balance of the equation. Dividing both sides by the same non-zero number ensures that the equality remains true.
  3. Perform the division: When we divide both sides of the equation by 2, we get (2x)/2 = 64/2. On the left side, the 2 in the numerator and the 2 in the denominator cancel each other out, leaving us with just x. On the right side, 64 divided by 2 equals 32.
  4. Simplify the equation: After performing the division, the equation simplifies to x = 32. This result tells us that the value of x that satisfies the original equation is 32. We have successfully isolated x and found its value.
  5. Verify the solution: To ensure the accuracy of our solution, we can substitute the value of x back into the original equation. If we replace x with 32 in the equation 2x = 64, we get 2(32) = 64. Performing the multiplication, we find that 64 = 64, which is a true statement. This verification step confirms that our solution of x = 32 is correct.

Through these steps, we have demonstrated a systematic approach to solving for x in a linear equation. This method of isolating the variable by performing inverse operations is a fundamental technique in algebra and is applicable to a wide range of equations. Understanding and mastering this process is essential for success in more advanced mathematical concepts.

Evaluating Expressions Using the Value of x

Now that we have successfully determined the value of x to be 32, we can proceed to the next phase of our mathematical journey: evaluating various expressions that involve x. This exercise is not just about substituting a numerical value into a formula; it's about understanding how different mathematical operations interact and how the value of a variable can impact the outcome of an expression. We will be working with a diverse set of expressions, ranging from simple linear expressions to more complex exponential expressions, each requiring a slightly different approach. This section will provide a detailed, step-by-step guide to evaluating each expression, reinforcing your understanding of mathematical operations and algebraic principles. Let's dive in and explore how the value of x shapes the results of these expressions.

a. Evaluating 3x

In this part, our objective is to evaluate the expression 3x. This is a straightforward linear expression, where x is multiplied by the coefficient 3. To find the value of 3x, we simply need to substitute the value of x, which we previously determined to be 32, into the expression. This process of substitution is a fundamental technique in algebra and allows us to determine the numerical value of an expression when we know the value of the variable. Let's break down the steps involved:

  1. Write down the expression: The expression we need to evaluate is 3x. This expression represents three times the value of x.
  2. Substitute the value of x: We know that x = 32. So, we substitute 32 for x in the expression, which gives us 3(32). This step replaces the variable with its numerical value, allowing us to perform the arithmetic operation.
  3. Perform the multiplication: Now, we perform the multiplication 3 * 32. This is a basic arithmetic operation that can be done manually or with the help of a calculator. The result of this multiplication is the numerical value of the expression.
  4. Determine the result: Multiplying 3 by 32, we get 96. Therefore, the value of the expression 3x when x = 32 is 96. This result tells us the specific numerical value of the expression for the given value of the variable.

Through this process, we have successfully evaluated the linear expression 3x. This simple example demonstrates the power of substitution in algebra and how it allows us to bridge the gap between symbolic expressions and numerical values. Understanding this process is crucial for tackling more complex expressions and equations.

b. Evaluating 2x - 3

Next, we will tackle the expression 2x - 3. This expression combines both multiplication and subtraction, adding a slight layer of complexity compared to the previous example. To evaluate this expression, we again need to substitute the value of x, which is 32, into the expression. However, we must also adhere to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order dictates the sequence in which we perform mathematical operations to ensure we arrive at the correct result. In this case, we will perform the multiplication before the subtraction. Let's outline the steps:

  1. Write down the expression: The expression we are working with is 2x - 3. This expression represents two times the value of x, minus 3.
  2. Substitute the value of x: We substitute x = 32 into the expression, resulting in 2(32) - 3. This step replaces the variable with its numerical value, setting the stage for the arithmetic operations.
  3. Perform the multiplication: According to the order of operations, we perform the multiplication first. So, we multiply 2 by 32, which gives us 64. The expression now becomes 64 - 3.
  4. Perform the subtraction: Next, we perform the subtraction. We subtract 3 from 64, which results in 61.
  5. Determine the result: Therefore, the value of the expression 2x - 3 when x = 32 is 61. This is the final numerical value of the expression for the given value of the variable.

By following the order of operations, we have accurately evaluated the expression 2x - 3. This example reinforces the importance of adhering to mathematical conventions to ensure consistent and correct results. It also demonstrates how expressions can combine different operations and how we can systematically work through them to find the final value.

c. Evaluating 4^(2x - 3)

Now we move on to a more complex expression: 4^(2x - 3). This expression introduces an exponent, which means we are dealing with exponential notation. Evaluating exponential expressions requires careful attention to the order of operations and a solid understanding of how exponents work. The base of the exponent is 4, and the exponent itself is the expression (2x - 3). This means we first need to evaluate the expression in the exponent before we can raise 4 to that power. Again, we will substitute the value of x, which is 32, into the expression and follow the order of operations. Let's break down the process:

  1. Write down the expression: The expression we need to evaluate is 4^(2x - 3). This expression represents 4 raised to the power of (2x - 3).
  2. Substitute the value of x: We substitute x = 32 into the expression, resulting in 4^(2(32) - 3). This step replaces the variable with its numerical value, setting up the calculation of the exponent.
  3. Evaluate the exponent: We need to evaluate the expression (2(32) - 3) first. Following the order of operations, we perform the multiplication before the subtraction. So, we multiply 2 by 32, which gives us 64. The expression now becomes 4^(64 - 3).
  4. Simplify the exponent: Next, we perform the subtraction in the exponent. We subtract 3 from 64, which results in 61. The expression now becomes 4^61.
  5. Calculate the exponential value: Finally, we need to calculate 4^61. This means we are raising 4 to the power of 61, which is a very large number. This calculation is best done using a calculator with exponential capabilities.
  6. Determine the result: Using a calculator, we find that 4^61 is approximately 1.152921504606846976 * 10^37. This is a very large number, demonstrating the rapid growth that can occur with exponential expressions.

Through this process, we have successfully evaluated the exponential expression 4^(2x - 3). This example highlights the importance of the order of operations when dealing with exponents and the significant impact that exponents can have on the magnitude of a number. It also demonstrates the utility of calculators in handling large numbers and complex calculations.

d. Evaluating (-1/3)^(-2x)

Lastly, we will evaluate the expression (-1/3)^(-2x). This expression introduces two new concepts: a negative fraction as the base and a negative exponent. Working with negative exponents and fractional bases requires a solid understanding of exponent rules and fraction manipulation. A negative exponent indicates that we need to take the reciprocal of the base and then raise it to the positive version of the exponent. The fractional base adds another layer of complexity, as we need to apply the exponent to both the numerator and the denominator. As before, we will substitute the value of x, which is 32, into the expression and follow the order of operations. Let's break down the steps:

  1. Write down the expression: The expression we need to evaluate is (-1/3)^(-2x). This expression represents the fraction -1/3 raised to the power of -2x.
  2. Substitute the value of x: We substitute x = 32 into the expression, resulting in (-1/3)^(-2(32)). This step replaces the variable with its numerical value, setting up the calculation of the exponent.
  3. Simplify the exponent: We need to simplify the exponent -2(32). Multiplying -2 by 32, we get -64. The expression now becomes (-1/3)^(-64).
  4. Apply the negative exponent rule: A negative exponent means we need to take the reciprocal of the base and change the sign of the exponent. The reciprocal of -1/3 is -3/1, which is simply -3. So, the expression becomes (-3)^64.
  5. Calculate the exponential value: Now, we need to calculate (-3)^64. This means we are raising -3 to the power of 64. Since the exponent is an even number, the result will be positive. This calculation is best done using a calculator with exponential capabilities.
  6. Determine the result: Using a calculator, we find that (-3)^64 is approximately 3.4336838202925124846578490892804 * 10^30. This is an extremely large positive number, highlighting the impact of a large even exponent on a negative base.

Through this process, we have successfully evaluated the expression (-1/3)^(-2x). This example demonstrates how to work with negative exponents and fractional bases, reinforcing the importance of exponent rules in algebraic manipulations. It also showcases the potential for very large numbers to arise from exponential expressions, especially when dealing with large exponents.

In this comprehensive exploration, we have successfully navigated the process of solving for x in the equation 2x = 64 and subsequently applied this knowledge to evaluate a series of diverse expressions. We began by isolating x through the fundamental principle of maintaining equation balance, ultimately determining that x = 32. This foundational step paved the way for the subsequent evaluations.

We then embarked on evaluating four distinct expressions, each presenting its unique challenges and reinforcing different mathematical concepts. We tackled the linear expression 3x, the combined multiplication and subtraction expression 2x - 3, the exponential expression 4^(2x - 3), and the complex expression (-1/3)^(-2x), which involved a negative fractional base and a negative exponent. Each evaluation required careful application of the order of operations and a solid understanding of algebraic rules and exponent properties.

Through these exercises, we have not only honed our algebraic skills but also gained a deeper appreciation for the interconnectedness of mathematical concepts. We have seen how the value of a single variable can influence the outcome of various expressions and how different mathematical operations interact to shape the final result. This journey serves as a testament to the power of systematic problem-solving and the importance of mastering fundamental algebraic principles. Whether you are a student seeking to strengthen your understanding or an enthusiast looking to refresh your knowledge, the skills and insights gained from this exploration will undoubtedly prove valuable in your mathematical endeavors.