Evaluate The Expression √[3]{-216} + √{25} - [ 5 × (-3)² ÷ (6 - 21) ]
Let's embark on a mathematical journey to evaluate the expression √[3]{-216} + √{25} - [ 5 × (-3)² ÷ (6 - 21) ]. This seemingly complex expression can be simplified by meticulously following the order of operations and applying the fundamental principles of arithmetic. This article aims to dissect this expression step by step, ensuring clarity and understanding for anyone looking to enhance their mathematical prowess. Understanding and mastering such evaluations not only solidifies your foundational mathematical skills but also equips you to tackle more intricate problems in algebra and beyond. The process involves dealing with cube roots, square roots, exponents, multiplication, division, addition, and subtraction, all within a single expression. By breaking down each component and operation, we will arrive at the final solution with confidence and precision. This exercise is a quintessential example of how complex mathematical expressions can be simplified through methodical application of mathematical rules and principles. Join us as we unravel this expression, turning complexity into clarity and boosting your confidence in mathematical evaluations.
Breaking Down the Expression
The expression we aim to evaluate the expression is: √[3]{-216} + √{25} - [ 5 × (-3)² ÷ (6 - 21) ]. To solve this, we'll break it down into smaller, manageable parts. The key to solving any mathematical expression lies in understanding the order of operations, often remembered by the acronym PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). This order ensures that we tackle each operation in the correct sequence, leading to the accurate result. Our expression includes cube roots, square roots, exponents, parentheses, multiplication, division, subtraction, and addition, making it a comprehensive exercise in applying the order of operations. By systematically addressing each component, we can transform the complex into the simple. Let's begin by focusing on the roots and exponents, as they form the foundation for further calculations. By mastering the decomposition of such expressions, one gains not only the ability to solve problems but also a deeper appreciation for the elegance and structure inherent in mathematics. This initial breakdown sets the stage for a smooth and efficient solution, making the entire process more understandable and less intimidating.
1. Evaluating the Cube Root: √[3]{-216}
The first part of our expression involves finding the cube root of -216. Evaluating the cube root of a number means finding a value that, when multiplied by itself three times, equals the original number. In this case, we need to find a number 'x' such that x * x * x = -216. Since we are dealing with a negative number, we know the cube root will also be negative. The cube root of 216 is 6, as 6 * 6 * 6 = 216. Therefore, the cube root of -216 is -6. This step is crucial as it simplifies a complex operation into a single, manageable number. Understanding cube roots is fundamental in algebra and calculus, as they frequently appear in various mathematical contexts. Recognizing perfect cubes, such as 216, streamlines the evaluation process, saving time and reducing the likelihood of errors. This initial step not only provides a numerical value but also sets the tone for a methodical approach to the rest of the expression. Mastering the evaluation of roots and exponents is a cornerstone of mathematical proficiency, enabling one to confidently navigate more complex problems.
2. Calculating the Square Root: √{25}
Next, we need to calculate the square root of 25. Calculating the square root of a number involves finding a value that, when multiplied by itself, equals the original number. In other words, we are looking for a number 'y' such that y * y = 25. The square root of 25 is 5 because 5 multiplied by itself (5 * 5) equals 25. This is a straightforward calculation, but it’s important to remember that square roots always yield a non-negative result. Understanding square roots is essential in various mathematical fields, including geometry and algebra. This step further simplifies our expression, replacing a root operation with a single numerical value. The ability to quickly recognize and calculate square roots of perfect squares is a valuable skill in mathematical problem-solving. By accurately evaluating the square root of 25, we are one step closer to solving the entire expression. This seemingly simple step contributes significantly to the overall clarity and efficiency of the evaluation process, highlighting the importance of mastering basic mathematical operations.
3. Simplifying the Parenthetical Expression: (6 - 21)
Now, let's focus on the expression within the parentheses: (6 - 21). Simplifying the parenthetical expression involves performing the subtraction operation. Subtracting 21 from 6 gives us -15. This step is crucial because, according to the order of operations (PEMDAS/BODMAS), expressions within parentheses must be evaluated first. This result, -15, will be used in the subsequent division operation within the brackets. Parentheses play a pivotal role in defining the order of operations, ensuring that calculations are performed in the correct sequence. Mastering the simplification of expressions within parentheses is a fundamental skill in algebra and arithmetic. By correctly evaluating (6 - 21), we reduce the complexity of the overall expression and pave the way for the next steps in the solution. This attention to detail and adherence to the order of operations are key to achieving an accurate final result.
4. Evaluating the Exponent: (-3)²
Moving on, we need to evaluate the exponent in the expression: (-3)². Evaluating the exponent means raising -3 to the power of 2, which is equivalent to multiplying -3 by itself. So, (-3)² = (-3) * (-3) = 9. Remember that when a negative number is raised to an even power, the result is positive. Understanding exponents is crucial in various mathematical contexts, including algebra, calculus, and beyond. This step simplifies the expression further, replacing an exponential operation with a single numerical value. The ability to correctly evaluate exponents is a fundamental skill, preventing errors and ensuring accuracy in mathematical calculations. By finding that (-3)² equals 9, we continue to unravel the complexity of the original expression, bringing us closer to the final solution. This meticulous approach to each operation underscores the importance of mastering basic mathematical principles.
5. Performing Multiplication and Division: 5 × 9 ÷ (-15)
Next, we need to address the multiplication and division within the brackets: 5 × (-3)² ÷ (6 - 21), which simplifies to 5 × 9 ÷ (-15). Performing multiplication and division should be done from left to right. First, we multiply 5 by 9, which equals 45. Then, we divide 45 by -15. Dividing 45 by -15 gives us -3. This result is crucial as it simplifies the bracketed expression into a single value. Multiplication and division are inverse operations, and their correct execution is essential in evaluating mathematical expressions. Adhering to the left-to-right rule ensures that we follow the proper order of operations. By accurately performing the multiplication and division, we reduce the complexity of the expression and pave the way for the final addition and subtraction steps. This step demonstrates the importance of a methodical approach to mathematical problem-solving, where each operation is performed with precision.
Combining the Results
Now that we've simplified each part of the expression, let's combine the results. We have: √[3]{-216} = -6, √{25} = 5, and [ 5 × (-3)² ÷ (6 - 21) ] = -3. Our original expression was √[3]{-216} + √{25} - [ 5 × (-3)² ÷ (6 - 21) ]. Substituting the simplified values, we get -6 + 5 - (-3). Combining the results involves performing the addition and subtraction operations in the correct order. Remember that subtracting a negative number is the same as adding its positive counterpart. So, -6 + 5 - (-3) becomes -6 + 5 + 3. This step is crucial as it brings together all the individual calculations to arrive at the final solution. By carefully combining the results, we ensure that we have accounted for every operation and simplification performed earlier. This synthesis of intermediate values is a hallmark of effective mathematical problem-solving, demonstrating a clear and organized approach to complexity. The ability to accurately combine results is the final step in unlocking the answer to the expression.
6. Final Calculation: -6 + 5 + 3
Finally, let's perform the final calculation: -6 + 5 + 3. Final calculation involves adding and subtracting the numbers from left to right. First, we add -6 and 5, which gives us -1. Then, we add -1 and 3, which gives us 2. Therefore, the final result of the expression is 2. This calculation represents the culmination of all the previous steps, each carefully executed to simplify the original expression. Accuracy in this final stage is paramount, as any error here would negate the efforts made in the earlier steps. The result, 2, is the definitive answer to the problem, showcasing the power of methodical mathematical problem-solving. This final step underscores the importance of attention to detail and precision in mathematical calculations, ensuring that the solution is both correct and reliable.
Answer
Thus, √[3]{-216} + √{25} - [ 5 × (-3)² ÷ (6 - 21) ] = 2. However, based on the options provided (a. 18, b. 16, c. 15, d. 10, e. 12), it seems there might be a mistake in the calculation or the options. Let's re-evaluate the steps to ensure accuracy.
Re-evaluating the Calculation
Let's meticulously re-examine each step to pinpoint any potential errors. Re-evaluating the calculation is a critical part of problem-solving, ensuring that the final answer is accurate and reliable. This process involves retracing each step, verifying the operations, and checking the intermediate results. It's a methodical approach that helps identify and correct any mistakes that might have occurred. In the context of complex expressions, re-evaluation is not just a formality but a necessity. It ensures that the final answer aligns with the problem's requirements and that all mathematical principles have been correctly applied. By revisiting each operation, we can gain confidence in the correctness of the solution or identify areas that need further attention. This practice reinforces the importance of precision and attention to detail in mathematics.
Step 1: √[3]{-216} = -6 (Correct)
The cube root of -216 is indeed -6, as -6 multiplied by itself three times (-6 * -6 * -6) equals -216. This calculation is accurate and forms a solid foundation for the rest of the expression. Step 1: √[3]{-216} = -6 serves as a crucial starting point, simplifying a complex root operation into a manageable numerical value. Verifying this step reaffirms the correctness of the initial assessment and ensures that subsequent calculations are based on a sound foundation. The precision in evaluating the cube root is essential, as it directly impacts the final result. By confirming the accuracy of this step, we move forward with confidence, knowing that the foundational element of the expression is correctly addressed. This meticulous approach to each step highlights the importance of verification in mathematical problem-solving.
Step 2: √{25} = 5 (Correct)
The square root of 25 is indeed 5, as 5 multiplied by itself (5 * 5) equals 25. This is a straightforward calculation and is correctly evaluated. Step 2: √{25} = 5 further simplifies the expression, replacing another root operation with a numerical value. Confirming the accuracy of this step is essential, as it builds upon the foundation established in the previous step. The correct evaluation of the square root is pivotal in ensuring the overall accuracy of the solution. By verifying this calculation, we reinforce the importance of precision and attention to detail in mathematical problem-solving. This step-by-step approach allows for a clear and methodical progression towards the final answer.
Step 3: (6 - 21) = -15 (Correct)
The expression within the parentheses, (6 - 21), correctly simplifies to -15. This step adheres to the order of operations, ensuring that parenthetical expressions are evaluated first. Step 3: (6 - 21) = -15 is a crucial simplification that sets the stage for subsequent calculations within the brackets. Verifying the accuracy of this step is paramount, as it directly impacts the overall solution. The correct execution of this subtraction operation is essential in maintaining the integrity of the mathematical expression. By confirming this step, we reinforce the importance of following the order of operations and performing calculations with precision.
Step 4: (-3)² = 9 (Correct)
Evaluating the exponent, (-3)², correctly results in 9. This is because -3 multiplied by itself (-3 * -3) equals 9. Step 4: (-3)² = 9 is a critical step in simplifying the expression, replacing an exponential operation with a numerical value. Verifying the accuracy of this step is essential, as it directly influences the subsequent multiplication and division operations. The correct evaluation of the exponent is fundamental in ensuring the overall accuracy of the solution. By confirming this step, we highlight the importance of understanding exponential operations and performing calculations with precision.
Step 5: 5 × 9 ÷ (-15) = -3 (Correct)
Performing the multiplication and division from left to right, 5 × 9 ÷ (-15) correctly simplifies to -3. First, 5 multiplied by 9 equals 45. Then, 45 divided by -15 equals -3. Step 5: 5 × 9 ÷ (-15) = -3 is a crucial simplification that reduces the bracketed expression to a single numerical value. Verifying the accuracy of this step is paramount, as it directly impacts the final calculation. The correct execution of these operations, following the order of operations, is essential in maintaining the integrity of the mathematical expression. By confirming this step, we reinforce the importance of precision and attention to detail in mathematical problem-solving.
Step 6: -6 + 5 - (-3) = -6 + 5 + 3 = 2 (Correct)
Combining the results, -6 + 5 - (-3) correctly simplifies to 2. This involves adding and subtracting the numbers from left to right, remembering that subtracting a negative number is the same as adding its positive counterpart. Step 6: -6 + 5 - (-3) = -6 + 5 + 3 = 2 represents the final calculation, bringing together all the simplified values to arrive at the final result. Verifying the accuracy of this step is essential, as it confirms the overall correctness of the solution. The meticulous execution of these addition and subtraction operations is crucial in ensuring the reliability of the answer. By confirming this step, we highlight the importance of precision in mathematical calculations.
Conclusion: Re-evaluation Confirmed Accuracy
Upon re-evaluating each step, we confirm that our calculation is accurate. The expression √[3]{-216} + √{25} - [ 5 × (-3)² ÷ (6 - 21) ] indeed equals 2. Conclusion: Re-evaluation Confirmed Accuracy underscores the importance of methodical verification in mathematical problem-solving. Despite the initial absence of the correct answer in the provided options, the re-evaluation process affirms the accuracy of the derived solution. This process highlights the value of precision, attention to detail, and adherence to the order of operations in mathematical calculations. The ability to retrace steps and verify results is a crucial skill in mathematics, ensuring that the final answer is both reliable and correct. This meticulous approach not only confirms the solution but also enhances confidence in one's problem-solving abilities.
Final Answer
Based on our calculations, the correct answer is 2, which is not among the options provided. There might be an error in the given options. This underscores the importance of double-checking both the calculations and the options when solving mathematical problems.
