Solving -4√[3]{-6m^2} When M=6 A Step-by-Step Guide

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In the realm of mathematics, expressions often present themselves as puzzles, challenging us to unravel their hidden values. Today, we embark on a journey to decipher the expression $-4 \sqrt[3]{-6m^2}$ when the variable m takes on the value of 6. This seemingly complex expression can be tamed with a systematic approach, and we'll guide you through each step to arrive at the correct answer.

Delving into the Expression: A Step-by-Step Solution

To solve this mathematical puzzle, we'll meticulously follow the order of operations, a fundamental principle in mathematics that dictates the sequence in which operations should be performed. This ensures we arrive at the accurate result.

1. Substitution: Replacing 'm' with its Value

The first step in our journey is to replace the variable m with its given value, which is 6. This substitution transforms the expression into a purely numerical one, paving the way for further simplification. So, we substitute m = 6 into the expression:

$-4 \sqrt[3]{-6m^2} = -4 \sqrt[3]{-6(6)^2}$

2. Exponentiation: Squaring the Value of 6

Next, we encounter an exponent, the superscript 2 attached to the number 6. This signifies that we need to square the value of 6, which means multiplying it by itself:

6² = 6 * 6 = 36

Now, our expression transforms further:

$-4 \sqrt[3]{-6(6)^2} = -4 \sqrt[3]{-6 * 36}$

3. Multiplication: Multiplying -6 by 36

Moving along, we encounter a multiplication operation within the cube root. We need to multiply -6 by 36:

-6 * 36 = -216

Our expression now takes the form:

$-4 \sqrt[3]{-6 * 36} = -4 \sqrt[3]{-216}$

4. Cube Root: Finding the Cube Root of -216

The heart of our expression lies within the cube root, symbolized by the radical sign with a small 3 as its index. This operation asks us to find a number that, when multiplied by itself three times, equals -216. In this case, the cube root of -216 is -6:

$\sqrt[3]{-216} = -6$

Our expression simplifies further:

$-4 \sqrt[3]{-216} = -4 * (-6)$

5. Final Multiplication: Multiplying -4 by -6

Finally, we arrive at the last operation, a simple multiplication. We need to multiply -4 by -6. Remember, when multiplying two negative numbers, the result is positive:

-4 * (-6) = 24

Thus, the final value of the expression is:

$-4 \sqrt[3]{-6m^2} = 24$

The Answer Revealed: Option B, 24

After meticulously navigating through the steps of substitution, exponentiation, multiplication, and cube root extraction, we arrive at the solution: 24. This corresponds to option B in the given choices.

Therefore, the correct answer is B. 24.

Key Concepts in Action: A Recap

This problem elegantly demonstrates the application of several key mathematical concepts:

• Order of Operations: The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is crucial for consistent and accurate mathematical calculations. We followed this order meticulously to solve the expression.
• Substitution: Substituting variables with their given values is a fundamental technique in algebra, allowing us to transform expressions into numerical forms.
• Exponents: Exponents represent repeated multiplication, and understanding their properties is essential for simplifying expressions.
• Cube Roots: Cube roots are the inverse operation of cubing, and they help us find the number that, when multiplied by itself three times, yields a specific value.

Mastering Mathematical Expressions: A Path to Confidence

Mathematical expressions, like the one we've tackled today, often appear daunting at first glance. However, by breaking them down into smaller, manageable steps and applying the fundamental principles of mathematics, we can unravel their complexities and arrive at the correct solutions. Remember, practice is key to mastering these skills and building confidence in your mathematical abilities.

In conclusion, the value of the expression $-4 \sqrt[3]{-6m^2}$ when m=6 is 24, which corresponds to option B. This problem showcases the power of systematic problem-solving and the importance of understanding fundamental mathematical concepts.

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Mathematical expressions can sometimes appear intricate and challenging, but with a methodical approach and a firm grasp of fundamental concepts, we can unravel their complexities. In this article, we will dissect the expression $-4 \sqrt[3]{-6m^2}$ when $m = 6$, providing a step-by-step solution and highlighting the key mathematical principles involved.

Unveiling the Solution: A Step-by-Step Guide

To accurately evaluate the expression $-4 \sqrt[3]{-6m^2}$ when $m = 6$, we must adhere to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This ensures that we perform the operations in the correct sequence, leading us to the accurate result.

Step 1: Substitution

The initial step involves replacing the variable m with its given value, which is 6. This substitution transforms the expression from an algebraic form to a numerical one, making it easier to evaluate.

Substituting $m = 6$ into the expression, we get:

$-4 \sqrt[3]{-6m^2} = -4 \sqrt[3]{-6(6)^2}$

Step 2: Exponentiation

Next, we encounter an exponent, the superscript 2, which indicates that we need to square the value of 6. Squaring a number means multiplying it by itself.

Calculating $6^2$, we have:

$6^2 = 6 * 6 = 36$

Our expression now becomes:

$-4 \sqrt[3]{-6(6)^2} = -4 \sqrt[3]{-6 * 36}$

Step 3: Multiplication

Within the cube root, we have a multiplication operation. We need to multiply -6 by 36.

Performing the multiplication:

$-6 * 36 = -216$

The expression now simplifies to:

$-4 \sqrt[3]{-6 * 36} = -4 \sqrt[3]{-216}$

Step 4: Cube Root

The cube root operation, denoted by the radical symbol with an index of 3, asks us to find a number that, when multiplied by itself three times, equals the radicand (the number under the radical). In this case, we need to find the cube root of -216.

The cube root of -216 is -6, because:

$(-6) * (-6) * (-6) = -216$

Therefore,

$\sqrt[3]{-216} = -6$

Our expression now becomes:

$-4 \sqrt[3]{-216} = -4 * (-6)$

Step 5: Final Multiplication

The final step involves multiplying -4 by -6. Remember that the product of two negative numbers is positive.

Performing the multiplication:

$-4 * (-6) = 24$

Therefore, the value of the expression is:

$-4 \sqrt[3]{-6m^2} = 24$

The Solution: Option B, 24

By carefully following the order of operations and performing each step meticulously, we have arrived at the solution: 24. This corresponds to option B in the given multiple-choice options.

Thus, the correct answer is B. 24.

Core Mathematical Concepts: A Review

This problem effectively illustrates the application of several fundamental mathematical concepts:

• Order of Operations (PEMDAS): The order of operations is a cornerstone of mathematical calculations, ensuring that we perform operations in the correct sequence to arrive at accurate results. We meticulously followed PEMDAS throughout the solution process.
• Substitution: Substituting variables with their given values is a fundamental technique in algebra, allowing us to transform expressions into numerical forms that can be easily evaluated.
• Exponents: Exponents represent repeated multiplication and play a crucial role in simplifying expressions. We used the concept of squaring to evaluate $6^2$.
• Cube Roots: Cube roots are the inverse operation of cubing and help us find the number that, when multiplied by itself three times, yields a specific value. We successfully calculated the cube root of -216.

Building Confidence in Mathematical Problem-Solving

Mathematical expressions may initially seem intimidating, but by breaking them down into smaller, more manageable steps and applying core mathematical principles, we can conquer their complexities and find the correct solutions. Consistent practice is the key to developing proficiency and confidence in your mathematical abilities.

In conclusion, the value of the expression $-4 \sqrt[3]{-6m^2}$ when $m = 6$ is 24, which corresponds to option B. This problem demonstrates the power of a systematic approach and the importance of mastering fundamental mathematical concepts.

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Mathematical expressions can often seem like cryptic codes, but with the right tools and techniques, we can decipher their meaning and arrive at the correct solutions. In this article, we'll unravel the expression $-4 \sqrt[3]{-6m^2}$ when m equals 6, guiding you through each step of the process and highlighting the key mathematical principles at play.

Embarking on the Solution Journey: A Step-by-Step Approach

To solve this mathematical puzzle, we'll employ a systematic approach, adhering to the order of operations – a fundamental principle in mathematics that dictates the sequence in which operations should be performed. This ensures we reach the accurate result.

1. Initial Substitution: Replacing 'm' with 6

Our first step is to replace the variable m with its designated value, which is 6. This substitution transforms the expression into a purely numerical one, setting the stage for subsequent simplifications. We substitute m = 6 into the expression:

$-4 \sqrt[3]{-6m^2} = -4 \sqrt[3]{-6(6)^2}$

2. Taming the Exponent: Squaring the Value of 6

Next, we encounter an exponent, the superscript 2 attached to the number 6. This signifies that we need to square the value of 6, meaning we multiply it by itself:

$6^2 = 6 * 6 = 36$

Our expression now evolves:

$-4 \sqrt[3]{-6(6)^2} = -4 \sqrt[3]{-6 * 36}$

3. Multiplication Within: Combining -6 and 36

Moving forward, we encounter a multiplication operation nestled within the cube root. We need to multiply -6 by 36:

-6 * 36 = -216

Our expression takes a new form:

$-4 \sqrt[3]{-6 * 36} = -4 \sqrt[3]{-216}$

4. Unveiling the Cube Root: Finding the Cube Root of -216

The core of our expression resides within the cube root, symbolized by the radical sign with a small 3 as its index. This operation prompts us to find a number that, when multiplied by itself three times, yields -216. In this instance, the cube root of -216 is -6:

$\sqrt[3]{-216} = -6$

Our expression simplifies further:

$-4 \sqrt[3]{-216} = -4 * (-6)$

5. The Grand Finale: Multiplying -4 by -6

Finally, we arrive at the concluding operation, a simple multiplication. We need to multiply -4 by -6. Remember, when multiplying two negative numbers, the result is positive:

-4 * (-6) = 24

Thus, the final value of the expression emerges:

$-4 \sqrt[3]{-6m^2} = 24$

Solution Achieved: Option B, 24

After diligently traversing through the steps of substitution, exponentiation, multiplication, and cube root extraction, we arrive at the solution: 24. This aligns with option B in the provided choices.

Therefore, the correct answer is B. 24.

Mathematical Pillars in Action: A Summary

This problem elegantly showcases the application of several fundamental mathematical concepts:

• The Order of Operations: The order of operations, often recalled by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is paramount for consistent and accurate mathematical calculations. We adhered to this order meticulously to solve the expression.
• Substitution Power: Substituting variables with their given values is a cornerstone technique in algebra, enabling us to transform expressions into numerical forms.
• Exponents Unveiled: Exponents signify repeated multiplication, and grasping their properties is vital for simplifying expressions.
• The Essence of Cube Roots: Cube roots serve as the inverse operation of cubing, assisting us in finding the number that, when multiplied by itself three times, produces a specific value.

Elevating Mathematical Proficiency: A Path to Confidence

Mathematical expressions, such as the one we've tackled today, often appear intimidating initially. However, by dissecting them into smaller, manageable steps and applying the fundamental principles of mathematics, we can conquer their complexities and attain accurate solutions. Remember, consistent practice is the key to honing these skills and building confidence in your mathematical prowess.

In conclusion, the value of the expression $-4 \sqrt[3]{-6m^2}$ when m=6 is 24, which corresponds to option B. This problem underscores the significance of systematic problem-solving and the mastery of fundamental mathematical concepts.