Evaluating (2⁰ + 4⁻¹/²)² A Step-by-Step Solution
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This article delves into the step-by-step evaluation of the mathematical expression (2⁰ + 4⁻¹/²)². We will explore the fundamental concepts of exponents, including zero exponents and negative exponents, and apply them to simplify the expression. This process will provide a comprehensive understanding of how to manipulate and solve mathematical expressions involving exponents and fractions. The solution will not only reveal the final answer but also illustrate the underlying principles that govern such calculations. Throughout this exploration, clarity and precision will be prioritized to ensure that the steps are easily understood and replicated. By the end of this article, readers will be equipped with the knowledge and skills to confidently tackle similar mathematical problems.
Understanding the Basics of Exponents
Before diving into the evaluation of the expression, it's crucial to understand the basic rules of exponents. An exponent indicates how many times a base number is multiplied by itself. For example, in the expression 2³, the base is 2, and the exponent is 3, meaning 2 is multiplied by itself three times (2 * 2 * 2 = 8). Zero exponents and negative exponents are particularly important in this context. Any non-zero number raised to the power of 0 is equal to 1. This is a fundamental rule that simplifies many mathematical expressions. In our case, 2⁰ will be simplified to 1. Negative exponents indicate the reciprocal of the base raised to the positive exponent. For example, x⁻ⁿ is equal to 1/xⁿ. This rule will be crucial in simplifying the term 4⁻¹/². Understanding these principles forms the foundation for effectively manipulating exponential expressions.
Zero Exponents and Their Significance
The concept of a zero exponent is a cornerstone of exponential notation. It might seem counterintuitive that any number raised to the power of zero equals one, but this rule is essential for maintaining consistency in mathematical operations. Consider the pattern of decreasing exponents: 2³, 2², 2¹, and then 2⁰. Following the pattern, each term is divided by 2 to get the next term: 2³ = 8, 2² = 4, 2¹ = 2. Continuing this pattern, 2⁰ should be 2/2, which equals 1. This logical progression demonstrates why any non-zero number raised to the power of zero is defined as 1. This rule is not just a mathematical convention; it has deep roots in the structure of exponential and logarithmic functions. In the given expression, 2⁰ is a critical component, and its simplification to 1 is the first step in solving the problem. Grasping the significance of zero exponents is vital for simplifying complex expressions and solving equations involving powers.
Negative Exponents and Reciprocals
Negative exponents introduce the concept of reciprocals into exponential expressions. A negative exponent indicates that the base should be taken to the reciprocal of its positive exponent. For instance, x⁻ⁿ is equivalent to 1/xⁿ. This rule is particularly useful when dealing with fractions and complex expressions. In the expression (2⁰ + 4⁻¹/²)², the term 4⁻¹/² involves a negative fractional exponent. To simplify this, we first recognize that the negative exponent means we will take the reciprocal. The fractional exponent 1/2 signifies the square root. Therefore, 4⁻¹/² is the reciprocal of the square root of 4. The square root of 4 is 2, and the reciprocal of 2 is 1/2. Thus, 4⁻¹/² simplifies to 1/2. Understanding how to handle negative exponents is essential for correctly evaluating expressions and solving equations. This concept allows for the seamless manipulation of exponential terms and is a fundamental aspect of algebra and calculus. The ability to convert negative exponents into positive exponents by taking reciprocals is a powerful tool in simplifying mathematical expressions.
Step-by-Step Evaluation of (2⁰ + 4⁻¹/²)²
Now, let's break down the evaluation of the expression (2⁰ + 4⁻¹/²)² step by step. This systematic approach ensures accuracy and clarity in the solution. Each step will be explained in detail, making the process easy to follow. The goal is to transform the complex expression into a simplified form by applying the rules of exponents and basic arithmetic operations. This methodical approach is applicable to a wide range of mathematical problems, providing a solid foundation for tackling more advanced concepts. The evaluation will proceed from the innermost operations to the outermost, adhering to the order of operations (PEMDAS/BODMAS). This structured methodology is essential for achieving the correct result and understanding the underlying mathematical principles.
Step 1: Simplify 2⁰
The first step in evaluating the expression is to simplify the term 2⁰. As established earlier, any non-zero number raised to the power of 0 is equal to 1. Therefore, 2⁰ simplifies to 1. This is a straightforward application of the zero exponent rule. Replacing 2⁰ with 1 simplifies the expression to (1 + 4⁻¹/²)². This substitution is crucial as it reduces the complexity of the expression and prepares it for the next steps. The simplicity of this step underscores the importance of understanding fundamental exponent rules. Correctly simplifying 2⁰ is the foundation for the subsequent calculations and is essential for arriving at the correct final answer. By applying this rule, we have taken the first significant step towards solving the problem.
Step 2: Simplify 4⁻¹/²
The next step is to simplify the term 4⁻¹/². This involves understanding both negative and fractional exponents. A negative exponent indicates that we need to take the reciprocal of the base raised to the positive exponent. So, 4⁻¹/² is equivalent to 1/(4¹/²). The fractional exponent 1/2 represents the square root. Therefore, 4¹/² is the square root of 4, which is 2. Substituting this back into our expression, we have 1/2. Thus, 4⁻¹/² simplifies to 1/2. This step demonstrates the combined application of negative and fractional exponent rules. By correctly simplifying 4⁻¹/², we transform the expression from (1 + 4⁻¹/²)² to (1 + 1/2)². This simplification is a key step in making the expression more manageable and easier to evaluate. The ability to handle fractional exponents and negative exponents is vital for advanced mathematical calculations.
Step 3: Add 1 and 1/2
Now that we have simplified 2⁰ to 1 and 4⁻¹/² to 1/2, the expression becomes (1 + 1/2)². The next step is to add 1 and 1/2. To add these numbers, we need to find a common denominator. In this case, the common denominator is 2. So, we rewrite 1 as 2/2. The expression then becomes (2/2 + 1/2)². Adding the fractions, we get (3/2)². This step involves basic arithmetic but is crucial for moving forward in the evaluation. The ability to add fractions correctly is fundamental to many mathematical operations. By adding 1 and 1/2, we further simplify the expression, bringing us closer to the final answer. This intermediate step showcases the importance of mastering basic mathematical skills in the context of more complex problems.
Step 4: Square 3/2
The final step in evaluating the expression is to square 3/2. Squaring a fraction means multiplying the fraction by itself. So, (3/2)² is equal to (3/2) * (3/2). To multiply fractions, we multiply the numerators (top numbers) together and the denominators (bottom numbers) together. Therefore, (3/2) * (3/2) equals (3 * 3) / (2 * 2), which simplifies to 9/4. This step completes the evaluation of the expression. We have systematically simplified the original expression (2⁰ + 4⁻¹/²)² to its final form, 9/4. This result demonstrates the power of applying exponent rules and basic arithmetic operations in a step-by-step manner. The final step of squaring the fraction solidifies the solution and provides the answer in its simplest form. The ability to perform this calculation accurately is a testament to understanding fractional arithmetic and exponentiation.
Final Answer
Therefore, the final answer to the expression (2⁰ + 4⁻¹/²)² is 9/4. This result was achieved through a series of systematic simplifications, applying the rules of exponents and basic arithmetic. Each step was carefully executed, ensuring accuracy and clarity in the solution. From simplifying the zero exponent to handling the negative fractional exponent, the process illustrates the importance of understanding fundamental mathematical principles. The final answer, 9/4, is a testament to the power of methodical problem-solving in mathematics. This comprehensive evaluation not only provides the correct answer but also reinforces the underlying concepts necessary for tackling similar mathematical challenges.
Conclusion
In conclusion, evaluating the expression (2⁰ + 4⁻¹/²)² involved a step-by-step process that highlighted the importance of understanding and applying exponent rules and basic arithmetic operations. The initial simplification of 2⁰ to 1 and 4⁻¹/² to 1/2 set the stage for the subsequent calculations. Adding 1 and 1/2 resulted in 3/2, and finally, squaring 3/2 yielded the final answer of 9/4. This exercise demonstrates how complex mathematical expressions can be simplified through a systematic approach. The solution reinforces the significance of mastering fundamental mathematical concepts, such as zero exponents, negative exponents, fractional exponents, and fraction arithmetic. By breaking down the problem into manageable steps, we were able to arrive at the correct answer with confidence. This approach is applicable to a wide range of mathematical problems and underscores the value of precision and clarity in mathematical problem-solving.
