If A = 2 And B + C = 2, Calculate (2a)^b * (2a)^c.

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Introduction

In this mathematical problem, we are given that a equals 2 and the sum of b and c equals 2. Our mission is to calculate the value of the expression (2a)^b multiplied by (2a)^c. This problem involves the fundamental principles of algebra and exponents. We will delve into the step-by-step solution, ensuring a clear and comprehensive understanding of each stage. The solution requires applying the properties of exponents and basic arithmetic operations. By carefully following the steps, we can accurately determine the final result. This exercise is not only a mathematical calculation but also an excellent way to reinforce our understanding of algebraic principles.

Understanding the Problem

To start, let's dissect the problem into smaller, manageable parts. We have three variables: a, b, and c. The value of a is explicitly given as 2. The relationship between b and c is defined by the equation b + c = 2. The expression we need to evaluate is (2a)^b * (2a)^c. This expression involves exponents, which means we need to understand how to manipulate them correctly. Exponents indicate the number of times a base is multiplied by itself. For instance, x^n means x multiplied by itself n times. A key property we'll use is the product of powers rule, which states that x^m * x^n = x^(m+n). This rule will help us simplify the given expression. By understanding these fundamental concepts, we set the stage for solving the problem efficiently and accurately.

Step-by-Step Solution

  1. Substitute the value of a: The first step is to replace a with its given value, which is 2. So, the expression (2a)^b * (2a)^c becomes (22)^b * (22)^c**, which simplifies to 4^b * 4^c.
  2. Apply the product of powers rule: Now, we can use the product of powers rule, which states that x^m * x^n = x^(m+n). Applying this rule to our expression, 4^b * 4^c becomes 4^(b+c). This step significantly simplifies the expression, making it easier to work with.
  3. Substitute the value of b + c: We know that b + c = 2. Substituting this value into our simplified expression, 4^(b+c) becomes 4^2. Now, we have a straightforward calculation to perform.
  4. Calculate the final value: Finally, we calculate 4^2, which means 4 multiplied by itself. So, 4^2 = 4 * 4 = 16. Therefore, the value of the expression (2a)^b * (2a)^c, given a = 2 and b + c = 2, is 16.

Detailed Explanation of Each Step

Let's delve deeper into each step to ensure clarity and comprehension.

  • Step 1: Substitute the value of a

    Substituting the value of a is a fundamental step in solving algebraic expressions. Given that a = 2, we replace every instance of a in the expression with 2. This transforms the expression (2a)^b * (2a)^c into (22)^b * (22)^c**. The multiplication within the parentheses is straightforward: 22 equals 4*. Thus, the expression further simplifies to 4^b * 4^c. This substitution is crucial as it eliminates one variable and sets the stage for applying exponent rules.

  • Step 2: Apply the product of powers rule

The product of powers rule is a key concept in exponents. This rule states that when multiplying two powers with the same base, you can add the exponents. Mathematically, this is expressed as x^m * x^n = x^(m+n). In our case, the base is 4, and the exponents are b and c. Applying the rule, 4^b * 4^c becomes 4^(b+c). This step is significant because it combines two terms into one, further simplifying the expression and bringing us closer to the final solution. The ability to recognize and apply this rule is essential for solving many algebraic problems involving exponents.

  • Step 3: Substitute the value of b + c

    We are given that b + c = 2. Substituting this value into our expression 4^(b+c), we replace (b+c) with 2. This transforms the expression into 4^2. This substitution is a direct application of the given information and is crucial for arriving at a numerical answer. At this point, the expression is significantly simplified, and we are left with a simple exponential term to evaluate. The process of substitution allows us to use known relationships between variables to simplify and solve expressions.

  • Step 4: Calculate the final value

    The final step involves calculating 4^2. This means 4 raised to the power of 2, which is equivalent to 4 multiplied by itself. So, 4^2 = 4 * 4 = 16. This calculation gives us the final value of the expression. Therefore, the solution to the problem is 16. This step demonstrates the practical application of exponents and basic multiplication, culminating in the final numerical answer. The accuracy of this step depends on the correct execution of the multiplication operation.

Conclusion

In conclusion, the value of the expression (2a)^b * (2a)^c, given a = 2 and b + c = 2, is 16. We arrived at this answer by systematically applying the principles of algebra and exponents. The step-by-step solution involved substituting the value of a, applying the product of powers rule, substituting the value of b + c, and finally calculating the result. Each step was crucial in simplifying the expression and arriving at the correct answer. This problem serves as a valuable exercise in understanding and applying fundamental algebraic concepts. By mastering these principles, we can confidently tackle more complex mathematical challenges. The process of breaking down the problem into smaller, manageable steps and applying the appropriate rules is a key strategy in problem-solving.

Final Answer

The final answer is 16.