Calculating 27^(2/3) A Step-by-Step Guide

In the realm of mathematics, exponential equations hold a significant place, and mastering them is crucial for various fields, including science, engineering, and finance. These equations involve exponents, which represent the power to which a base number is raised. In this comprehensive guide, we will delve into the intricacies of exponential equations, focusing on a specific example: calculating 27^(2/3). By understanding the underlying principles and applying the correct techniques, you will gain the ability to solve a wide range of exponential problems.

Before diving into the solution, let's establish a solid foundation by defining the core concepts. An exponent indicates how many times a base number is multiplied by itself. For instance, in the expression 2^3, 2 is the base, and 3 is the exponent, signifying that 2 is multiplied by itself three times (2 * 2 * 2 = 8). Fractional exponents, like the one in our problem, introduce an additional layer of complexity, as they represent both a root and a power.

A fractional exponent, such as a/b, can be interpreted as both a root and a power. The denominator (b) indicates the root to be taken, while the numerator (a) indicates the power to which the result is raised. In the case of 27^(2/3), the denominator 3 signifies the cube root, and the numerator 2 signifies the power to which the cube root is raised. This understanding is crucial for effectively solving exponential equations involving fractions. Mastering fractional exponents opens doors to tackling more intricate mathematical challenges.

Now, let's embark on a step-by-step journey to solve the equation 27^(2/3). This methodical approach will not only provide the solution but also reinforce the fundamental principles involved. By carefully dissecting the problem, we can gain a deeper understanding of the mathematical concepts at play.

Step 1: Calculate the Cube Root of 27

The first step involves calculating the cube root of 27. The cube root of a number is the value that, when multiplied by itself three times, equals the original number. In this case, we seek a number that, when cubed, equals 27. Through careful consideration or using a calculator, we can determine that the cube root of 27 is 3, as 3 * 3 * 3 = 27. This step is foundational for simplifying the expression and moving towards the final solution.

Step 2: Square the Result

Having found the cube root of 27 to be 3, the next step is to square this result. Squaring a number means multiplying it by itself. Therefore, we need to calculate 3 squared, which is 3 * 3 = 9. This step completes the process of applying the fractional exponent, as we have now both taken the root and raised the result to the appropriate power. The result of this step is the solution to the original equation.

Therefore, 27^(2/3) = 9.

Among the provided options, the correct answer is C. 9. This confirms that our step-by-step solution has led us to the accurate result. Selecting the correct answer reinforces the importance of meticulous calculations and a thorough understanding of the underlying mathematical principles. Accuracy is paramount in mathematics, and verifying the solution against the given options ensures that we have arrived at the correct conclusion.

Another effective approach to solving this problem involves prime factorization. Prime factorization is the process of breaking down a number into its prime factors, which are numbers divisible only by 1 and themselves. This method can provide a different perspective on the problem and further solidify our understanding of exponential equations.

Step 1: Prime Factorize 27

The prime factorization of 27 is 3 * 3 * 3, which can be written as 3^3. This representation highlights the fact that 27 is a perfect cube, which is crucial for simplifying the expression.

Step 2: Substitute the Prime Factorization into the Equation

Substituting 3^3 for 27 in the original equation, we get (3³⁾(2/3). This substitution allows us to apply the rules of exponents more directly.

Step 3: Apply the Power of a Power Rule

The power of a power rule states that (a^m)n = a^(m*n). Applying this rule, we multiply the exponents: 3 * (2/3) = 2. This simplifies the equation to 3^2.

Step 4: Calculate 3^2

Finally, we calculate 3^2, which is 3 * 3 = 9. This alternative method confirms our previous result and demonstrates the versatility of mathematical problem-solving techniques.

To effectively solve exponential equations, it is essential to grasp the key concepts and rules that govern them. These principles provide the framework for manipulating and simplifying expressions, leading to accurate solutions. Understanding these concepts not only aids in solving specific problems but also enhances overall mathematical proficiency.

• Definition of Exponents: An exponent indicates how many times a base number is multiplied by itself. For example, a^n means a multiplied by itself n times.
• Fractional Exponents: A fractional exponent a/b represents both a root and a power. The denominator b indicates the root, and the numerator a indicates the power.
• Power of a Power Rule: This rule states that (a^m)n = a^(m*n). It is crucial for simplifying expressions where an exponent is raised to another exponent.
• Prime Factorization: Breaking down a number into its prime factors can simplify calculations and provide a clearer understanding of the number's structure.

To solidify your understanding of exponential equations, practice is essential. Working through a variety of problems will help you internalize the concepts and develop problem-solving skills. Here are a few practice problems to get you started:

1. Calculate 8^(2/3)
2. Solve for x: 4^x = 16
3. Simplify: (2³⁾2
4. Evaluate: 16^(3/4)

By tackling these problems, you can reinforce your knowledge and build confidence in your ability to handle exponential equations.

In conclusion, solving exponential equations involves understanding the fundamental principles of exponents and applying the appropriate techniques. By breaking down fractional exponents, utilizing prime factorization, and applying the power of a power rule, we can effectively solve a wide range of problems. The example of calculating 27^(2/3) illustrates a methodical approach that can be applied to other exponential equations. Consistent practice and a solid grasp of the core concepts are the keys to mastering this essential mathematical skill. Whether you're a student, a professional, or simply someone with a passion for mathematics, understanding exponential equations will undoubtedly enhance your analytical and problem-solving abilities. So, embrace the challenge, delve into the world of exponents, and unlock the power of mathematical understanding.