Evaluating Fractional Exponents A Step By Step Guide To Solving 144^(3/2)

Fractional exponents can seem daunting at first glance, but they are a powerful and elegant way to express both powers and roots in a single notation. This article will provide a comprehensive exploration of fractional exponents, how they work, and how to evaluate expressions involving them. We'll specifically focus on the expression 144^(3/2), breaking it down step-by-step to arrive at the correct solution. This detailed explanation will not only help you understand the mechanics of the problem but also enhance your overall grasp of exponential and radical concepts in mathematics. Understanding fractional exponents is crucial for success in algebra, calculus, and various fields of science and engineering, where they frequently appear in formulas and equations. By the end of this guide, you will have a clear understanding of how to manipulate and solve expressions with fractional exponents, empowering you to tackle similar problems with confidence.

The concept of exponents is fundamental in mathematics, representing repeated multiplication of a base number. For example, 2^3 means 2 multiplied by itself three times (2 * 2 * 2 = 8). However, what happens when the exponent is not a whole number but a fraction? That's where fractional exponents come into play. A fractional exponent combines the concepts of powers and roots. The denominator of the fraction indicates the type of root to be taken, while the numerator indicates the power to which the base is raised. This dual nature of fractional exponents makes them a versatile tool in simplifying and solving mathematical problems. For instance, x^(1/2) is equivalent to the square root of x, and x^(1/3) represents the cube root of x. When the fractional exponent has a numerator other than 1, such as x^(m/n), it means taking the nth root of x and then raising the result to the power of m, or vice versa. This equivalence is a key concept in manipulating fractional exponents and simplifying complex expressions.

When dealing with fractional exponents, it's essential to understand the relationship between exponents and radicals. The expression a^(m/n) can be interpreted in two equivalent ways: as the nth root of a raised to the power of m, or as a raised to the power of m and then taking the nth root. Mathematically, this is expressed as a^(m/n) = (√[n]a)^m = √n. This equivalence allows us to choose the most convenient order of operations when evaluating an expression. For example, if we have 8^(2/3), we can either find the cube root of 8 first (which is 2) and then square it (2^2 = 4), or we can square 8 first (8^2 = 64) and then find the cube root of 64 (√[3]64 = 4). Both methods yield the same result, but one might be easier to compute depending on the specific numbers involved. This flexibility is a valuable tool in simplifying complex expressions and avoiding large intermediate numbers. Understanding this equivalence is fundamental to mastering fractional exponents and their applications.

To effectively evaluate 144^(3/2), we need to break it down into its components and understand the meaning of each part. The base number is 144, and the exponent is 3/2. As we discussed, a fractional exponent indicates both a power and a root. In this case, the denominator of the fraction (2) tells us to take the square root, and the numerator (3) tells us to raise the result to the power of 3. Therefore, 144^(3/2) can be interpreted as either the square root of 144 raised to the power of 3, or 144 raised to the power of 3 and then taking the square root. Understanding these two equivalent interpretations is crucial for choosing the most efficient method of calculation. This initial deconstruction of the expression is the first step towards simplifying it and arriving at the final answer. By recognizing the individual components and their roles, we can approach the problem in a structured and logical manner.

Now, let's apply this understanding to our specific problem. The expression 144^(3/2) can be rewritten using the radical notation as (√144)^3 or √(144^3). Both forms are mathematically equivalent, but one might be easier to compute depending on the context. In this case, finding the square root of 144 first is the simpler approach, as it avoids dealing with a much larger number resulting from cubing 144 initially. The square root of 144 is a well-known value, which makes this path more straightforward. This choice of approach highlights the importance of strategically selecting the order of operations when working with fractional exponents. By recognizing the relationship between roots and powers, we can simplify the calculation process and reduce the risk of errors. This step-by-step approach demonstrates the power of breaking down complex expressions into manageable components.

To continue, we first evaluate the square root of 144. The square root of 144 is 12 because 12 multiplied by itself (12 * 12) equals 144. This is a fundamental square root that is often memorized, making the calculation quicker. However, even if it's not immediately obvious, one can find the square root through prime factorization or other methods. Knowing common square roots is a valuable skill in mathematics, as it allows for faster and more efficient problem-solving. Once we've determined that √144 = 12, we can substitute this value back into our expression. This substitution simplifies the original problem, transforming it into a more manageable form. The process of finding and substituting known values is a common strategy in simplifying mathematical expressions and solving equations. This step demonstrates the importance of mastering basic mathematical facts and operations to tackle more complex problems.

Having found that √144 = 12, we now need to raise this result to the power of 3. This means calculating 12^3, which is 12 multiplied by itself three times: 12 * 12 * 12. This step is a straightforward application of the definition of exponents. While it might seem like a simple calculation, it's crucial to perform it accurately to arrive at the correct final answer. Understanding the meaning of exponents and how to calculate powers is a fundamental skill in mathematics, essential for solving a wide range of problems. This specific calculation highlights the importance of being comfortable with multiplication and the order of operations. By breaking the problem down into smaller, manageable steps, we've made it easier to perform the necessary calculations and avoid errors.

Now, let's perform the multiplication. First, we multiply 12 by 12, which equals 144. Then, we multiply this result (144) by 12 again. The calculation 144 * 12 can be done manually or with a calculator. Breaking it down, we have 144 * 10 = 1440 and 144 * 2 = 288. Adding these together, we get 1440 + 288 = 1728. Therefore, 12^3 equals 1728. This calculation demonstrates the importance of careful and accurate arithmetic. Even if the initial steps of the problem are understood perfectly, an error in the final calculation can lead to an incorrect answer. This step-by-step breakdown of the multiplication process highlights the value of attention to detail in mathematical problem-solving. The ability to perform accurate calculations, whether manually or with a calculator, is a crucial skill for success in mathematics and related fields.

Thus, we have determined that (√144)^3 = 12^3 = 1728. This means that 144^(3/2) = 1728. This result is the final answer to our original problem. By following a logical and systematic approach, we have successfully evaluated the expression involving a fractional exponent. This process has demonstrated the importance of understanding the relationship between exponents and radicals, breaking down complex problems into simpler steps, and performing accurate calculations. This comprehensive solution not only provides the answer but also reinforces the underlying mathematical concepts. The ability to solve problems involving fractional exponents is a valuable skill, applicable in various areas of mathematics and science.

Now that we have calculated the value of 144^(3/2) to be 1728, we need to compare this result with the given answer choices to identify the correct one. The answer choices are typically presented in a multiple-choice format, and carefully examining each option is crucial to ensure the correct selection. This step highlights the importance of not only arriving at the correct solution but also accurately identifying it within the given context. Sometimes, the answer choices might be designed to be tricky, including common errors or misconceptions. Therefore, a thorough comparison is necessary to avoid making a mistake in the final step. This process of verification and comparison is a critical aspect of problem-solving in mathematics and other disciplines.

The provided answer choices are: A. 216 B. 1728 C. √[3]12 D. √[3]72

By comparing our calculated result (1728) with these options, we can clearly see that option B (1728) matches our solution. The other options (216, √[3]12, and √[3]72) are not equal to 1728, and we can confidently eliminate them. This comparison process reinforces the importance of accurate calculation and careful attention to detail. By correctly evaluating the expression and then systematically comparing the result with the answer choices, we can ensure that we select the correct option. This step also demonstrates the value of having a clear and organized approach to problem-solving, allowing for efficient identification of the correct answer.

Therefore, the equivalent expression to 144^(3/2) is indeed 1728. This conclusion is based on our step-by-step calculation and comparison with the provided answer choices. By understanding the concepts of fractional exponents, breaking down the problem into manageable steps, performing accurate calculations, and carefully examining the answer choices, we have successfully solved the problem. This process serves as a model for approaching similar mathematical problems, emphasizing the importance of both conceptual understanding and procedural proficiency. The ability to confidently solve problems involving fractional exponents is a valuable skill in mathematics and related fields, and this detailed explanation has hopefully provided a clear and comprehensive understanding of the process.

Working with fractional exponents can sometimes lead to errors if certain common pitfalls are not avoided. One frequent mistake is misunderstanding the order of operations. As we've seen, a fractional exponent like m/n indicates both a root (nth root) and a power (raising to the mth power). It's crucial to remember that a^(m/n) is equivalent to (√[n]a)^m or √n, and choosing the most convenient order can simplify the calculation. Failing to apply this equivalence correctly can lead to incorrect results. This emphasis on the order of operations highlights a fundamental principle in mathematics, where the sequence of steps can significantly impact the outcome. Understanding and adhering to the correct order of operations is essential for accuracy in mathematical problem-solving.

Another common error is misinterpreting the meaning of the numerator and denominator in the fractional exponent. The denominator represents the index of the root (e.g., square root, cube root), while the numerator represents the power to which the base or the root is raised. Confusing these roles can lead to incorrect calculations. For example, mistaking 144^(3/2) as the cube root of 144 squared instead of the square root of 144 cubed would result in a completely different answer. This potential for confusion underscores the importance of a solid understanding of the notation and definitions related to fractional exponents. A clear grasp of these fundamental concepts is essential for avoiding errors and confidently solving problems involving fractional exponents.

Additionally, students sometimes struggle with simplifying expressions involving negative fractional exponents. A negative exponent indicates the reciprocal of the base raised to the positive exponent. For example, a^(-m/n) is equal to 1/(a^(m/n)). Failing to apply this rule correctly can lead to significant errors in calculations. Negative exponents are a crucial concept in algebra and calculus, and a thorough understanding of their properties is essential. Recognizing and applying the rule for negative exponents is a key step in simplifying and evaluating expressions correctly. This rule is not only applicable to fractional exponents but also to integer exponents, highlighting the interconnectedness of mathematical concepts.

Finally, it's important to be careful with arithmetic errors when performing the calculations. Even if the concept and the steps are understood correctly, mistakes in multiplication, division, or exponentiation can lead to an incorrect final answer. As we saw in our example, calculating 12^3 requires accurate multiplication. A simple arithmetic error in this step would invalidate the entire solution. This emphasizes the importance of double-checking calculations and using a calculator or other tools to verify results, especially in complex problems. Attention to detail and careful execution are crucial for success in mathematical problem-solving.

In conclusion, understanding and mastering fractional exponents is essential for success in various areas of mathematics. By breaking down the concept into its fundamental components – the relationship between exponents and radicals, the meaning of the numerator and denominator in the fraction, and the rules for negative exponents – we can confidently tackle problems involving fractional exponents. This detailed exploration of 144^(3/2) has demonstrated a step-by-step approach to evaluating such expressions, highlighting the importance of accurate calculations, attention to detail, and a clear understanding of the underlying concepts. The ability to manipulate and solve expressions with fractional exponents is a valuable skill, applicable in algebra, calculus, and beyond. By avoiding common mistakes and practicing regularly, anyone can develop proficiency in this area and enhance their overall mathematical capabilities. Fractional exponents are not just an abstract concept; they are a powerful tool for simplifying and solving problems in diverse fields, making their mastery a worthwhile endeavor.