Solving For X In 2^2 × 2^x = 16 An Exponential Equation

In this article, we will delve into solving an exponential equation. Exponential equations are equations in which the variable appears in the exponent. These types of equations often appear in various fields of mathematics and science, making it essential to understand how to solve them. Our specific problem involves finding the value of x that satisfies the equation 2^2 × 2^x = 16. We will explore the fundamental properties of exponents and apply them systematically to isolate x and determine its value. By understanding the steps involved in solving this equation, you will be able to approach similar problems with confidence and a clear strategy. This skill is particularly valuable in algebra and calculus, where exponential functions play a crucial role. So, let's get started and unlock the value of x in this intriguing exponential equation.

Before we dive into the solution, let's briefly understand exponential equations. Exponential equations are algebraic equations where the variable is in the exponent. They take the general form a^x = b, where a is the base, x is the exponent, and b is the result. Solving these equations involves finding the value of x that satisfies the equation. The properties of exponents play a crucial role in simplifying and solving these equations. For instance, the rule a^(m+n) = a^m × a^n allows us to combine terms with the same base, while the rule (a^m)n = a^(m×n) helps us simplify powers raised to powers. Additionally, understanding the relationship between exponential and logarithmic forms is crucial. The equation a^x = b can be rewritten in logarithmic form as log_a(b) = x, where log_a represents the logarithm to the base a. This transformation is particularly useful when isolating the variable in the exponent. The base of the exponential function is also significant; if the base is a positive number not equal to 1, the exponential function is either strictly increasing (if the base is greater than 1) or strictly decreasing (if the base is between 0 and 1). Exponential equations are prevalent in various applications, including compound interest calculations, population growth models, and radioactive decay problems. Mastering the techniques to solve them is, therefore, an essential skill in both mathematics and real-world scenarios. As we tackle the equation 2^2 × 2^x = 16, we'll apply these principles to find the correct value of x.

Our problem is to solve for x in the equation 2^2 × 2^x = 16. This equation involves exponential terms, and our goal is to isolate x and find its value. We will use the properties of exponents to simplify the equation and then solve for x. The question presents us with multiple-choice options: A. 2, B. 3, C. 4, and D. 1. To find the correct answer, we need to systematically manipulate the equation and determine which of these values satisfies the original expression. This exercise is not just about finding the correct answer but also about understanding the process of solving exponential equations, which is a fundamental skill in algebra. By breaking down the problem step-by-step, we can clarify how to handle similar equations in the future. So, let's proceed with our solution strategy, applying the principles of exponents and algebraic manipulation to identify the precise value of x that makes the equation true. Through this process, we will reinforce our understanding of exponential equations and enhance our problem-solving capabilities.

To solve the equation 2^2 × 2^x = 16, we'll follow these steps:

1. Simplify the left side of the equation using the properties of exponents. Recall that when multiplying exponential terms with the same base, we add the exponents. So, 2^2 × 2^x can be simplified as 2^(2+x).

   • Step 1 Detailed Explanation: When multiplying powers with the same base, you add the exponents. This is a fundamental property of exponents, which is mathematically expressed as a^m × a^n = a^(m+n). In our case, the base is 2, and we are multiplying 2^2 by 2^x. Applying this rule, we add the exponents 2 and x, which gives us 2^(2+x). This simplification is crucial because it combines the two exponential terms on the left side into a single term, making the equation easier to manage. This step sets the stage for further simplification and ultimately allows us to isolate x. The power of this property lies in its ability to consolidate exponential expressions, which is a technique frequently used in solving various algebraic problems. This simplification is a key step towards our goal of solving for x. By reducing the complexity of the equation, we make it more approachable and easier to solve. Let's proceed with the next steps to find the value of x.

2. Rewrite the right side of the equation as a power of 2. We know that 16 is 2^4. So, we can rewrite the equation as 2^(2+x) = 2^4.

   • Step 2 Detailed Explanation: Expressing both sides of the equation with the same base is a pivotal step in solving exponential equations. The number 16 can be written as a power of 2, specifically 2^4. This is because 2 multiplied by itself four times (2 × 2 × 2 × 2) equals 16. By rewriting 16 as 2^4, we create a situation where both sides of the equation have the same base, which is 2. Now, our equation looks like 2^(2+x) = 2^4. This transformation is significant because if two powers with the same base are equal, their exponents must also be equal. This logical deduction forms the basis for the next step in solving the equation, where we will equate the exponents and solve for x. Recognizing and applying this property is a cornerstone of solving exponential equations. It allows us to move from comparing exponential expressions to comparing simpler algebraic expressions, which significantly simplifies the problem. We're now closer to finding the value of x.

3. Since the bases are equal, we can equate the exponents. This gives us the equation 2 + x = 4.

   • Step 3 Detailed Explanation: When the bases on both sides of an exponential equation are the same, the exponents must be equal for the equation to hold true. This is a direct consequence of the one-to-one property of exponential functions. In our equation, we have 2^(2+x) = 2^4. Since the base is 2 on both sides, we can equate the exponents, which means 2 + x must be equal to 4. This step transforms the exponential equation into a simple linear equation, making it much easier to solve. Equating exponents is a powerful technique in solving exponential equations because it allows us to bypass the complexities of dealing with exponential expressions directly. Instead, we can focus on the relationship between the exponents, which are often simpler algebraic terms. This transition is a key problem-solving strategy in mathematics, where complex problems are often simplified by breaking them down into more manageable parts. Now that we have the linear equation 2 + x = 4, solving for x becomes a straightforward algebraic task. This step brings us closer to the final answer.

4. Solve for x. Subtract 2 from both sides of the equation: x = 4 - 2, which gives us x = 2.

   • Step 4 Detailed Explanation: To isolate x in the equation 2 + x = 4, we need to subtract 2 from both sides of the equation. This is a fundamental algebraic operation that maintains the balance of the equation. Subtracting 2 from both sides gives us x = 4 - 2, which simplifies to x = 2. This step provides us with the solution to the equation. We have successfully found the value of x that satisfies the original exponential equation. The simplicity of this step highlights the effectiveness of the previous steps in transforming the complex exponential equation into a basic linear equation. Solving for x in this manner demonstrates a key principle in algebra: isolating the variable by performing inverse operations. This is a technique used across many different types of equations and is an essential skill for any mathematics student. With x = 2, we have now solved the equation. We can verify this solution by substituting x = 2 back into the original equation.

5. Therefore, the value of x that satisfies the equation is 2.

   • Final Answer: After simplifying the equation and isolating x, we found that x = 2. This value satisfies the original equation 2^2 × 2^x = 16. Let's verify this by substituting x = 2 back into the equation: 2^2 × 2^2 = 4 × 4 = 16, which confirms that our solution is correct. Thus, the value of x that solves the equation is indeed 2. This final step of verification is crucial in mathematics to ensure the accuracy of the solution. It provides a sense of confidence in the answer and helps to avoid errors. By going through the steps of simplification, equating exponents, and solving for x, we have demonstrated a clear and methodical approach to solving exponential equations. This method can be applied to similar problems, making it a valuable skill in algebra. The solution x = 2 is the correct answer to the problem.

The correct answer is A. 2.

In conclusion, solving for x in the exponential equation 2^2 × 2^x = 16 involves understanding and applying the properties of exponents. By simplifying the equation using exponent rules, equating exponents, and solving for x, we found that the correct value is x = 2. This problem highlights the importance of algebraic manipulation and the ability to break down complex equations into simpler, solvable steps. Mastering these techniques is crucial for success in algebra and related fields. The methodical approach used in this solution can be applied to a wide range of exponential equations, providing a solid foundation for further mathematical studies. Exponential equations, like the one we solved, are not just theoretical exercises; they appear in various real-world applications, including finance, physics, and engineering. Therefore, understanding how to solve them is a valuable skill. The process we followed – simplifying, equating, and solving – is a general strategy that can be adapted to different problems. Remember, the key to solving mathematical problems is a clear understanding of the underlying principles and a systematic approach. Keep practicing, and you'll become more proficient in solving exponential equations and other algebraic challenges.