Calculating (0.064)^(2/3) A Step-by-Step Guide

In the realm of mathematics, fractional exponents often present a captivating challenge. This article delves into the intricacies of evaluating the expression $(0.064)^{\frac{2}{3}}$, providing a step-by-step guide that not only elucidates the solution but also underscores the fundamental principles governing fractional exponents. Our exploration will weave through the conversion of decimals to fractions, the application of exponent rules, and the simplification of expressions, ultimately revealing the numerical value of the given expression. This comprehensive guide aims to empower readers with a deep understanding of fractional exponents, enabling them to tackle similar mathematical puzzles with confidence and precision.

Understanding Fractional Exponents

Fractional exponents, at first glance, might appear daunting, but they are simply a shorthand notation for expressing both powers and roots. At its core, a fractional exponent like $\frac{2}{3}$ signifies two operations: raising a number to a power (the numerator) and taking a root (the denominator). For instance, $x^{\frac{a}{b}}$ is equivalent to $\sqrt[b]{x^a}$, where 'a' represents the power and 'b' represents the root. This duality is the key to unlocking the secrets of fractional exponents.

In our specific case, $(0.064)^{\frac{2}{3}}$ translates to finding the cube root of 0.064 and then squaring the result. This understanding is crucial because it breaks down a seemingly complex problem into manageable steps. To effectively tackle this problem, we need to be adept at converting decimals to fractions, understanding the properties of exponents and roots, and simplifying expressions. This foundational knowledge will not only help us solve this particular problem but also equip us to handle a wide array of mathematical challenges involving fractional exponents.

The elegance of fractional exponents lies in their ability to seamlessly combine two mathematical operations into one concise notation. This not only simplifies mathematical expressions but also provides a powerful tool for solving equations and modeling real-world phenomena. For instance, in physics, fractional exponents are used to describe the relationship between quantities in various contexts, such as the scaling of physical properties with size. In finance, they appear in calculations involving compound interest and growth rates. Therefore, mastering the concept of fractional exponents is not just an academic exercise; it is a gateway to understanding and solving problems in diverse fields.

Step 1: Converting the Decimal to a Fraction

The initial hurdle in evaluating $(0.064)^{\frac{2}{3}}$ lies in the decimal form of the base, 0.064. To navigate this, we must first convert the decimal into its fractional equivalent. The decimal 0.064 represents sixty-four thousandths, which can be written as $\frac{64}{1000}$. This conversion is crucial because it allows us to work with integers, which are generally easier to manipulate when dealing with exponents and roots.

The fraction $\frac{64}{1000}$ can be further simplified by finding the greatest common divisor (GCD) of 64 and 1000. The GCD is the largest number that divides both 64 and 1000 without leaving a remainder. In this case, the GCD is 8. Dividing both the numerator and the denominator by 8, we get $\frac{64 \div 8}{1000 \div 8} = \frac{8}{125}$. This simplified fraction is much easier to work with and sets the stage for the next steps in our evaluation.

The importance of this step cannot be overstated. Converting decimals to fractions not only simplifies calculations but also provides a clearer understanding of the underlying mathematical structure. Fractions allow us to express numbers as ratios, which is particularly useful when dealing with exponents and roots. Furthermore, simplifying fractions to their lowest terms reduces the complexity of the numbers involved, making the subsequent steps less prone to errors. This meticulous approach to converting and simplifying numbers is a hallmark of sound mathematical practice and is essential for achieving accurate results.

Step 2: Applying the Fractional Exponent

With the base now expressed as the fraction $\frac{8}{125}$, we can proceed to apply the fractional exponent $\frac{2}{3}$. Recall that a fractional exponent indicates both a power and a root. In this case, $(0.064)^{\frac{2}{3}}$ is equivalent to $\left(\frac{8}{125}\right)^{\frac{2}{3}}$, which means we need to find the cube root of $\frac{8}{125}$ and then square the result. This understanding is crucial for breaking down the problem into smaller, more manageable parts.

We can apply the exponent to both the numerator and the denominator separately, using the rule $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$. Thus, $\left(\frac{8}{125}\right)^{\frac{2}{3}}$ becomes $\frac{8^{\frac{2}{3}}}{125^{\frac{2}{3}}}$. This separation allows us to deal with each part individually, simplifying the overall process.

Next, we interpret the fractional exponent as a combination of a root and a power. The denominator of the fraction (3) indicates the root to be taken, and the numerator (2) indicates the power to which the result should be raised. Therefore, $8^{\frac{2}{3}}$ means taking the cube root of 8 and then squaring the result, while $125^{\frac{2}{3}}$ means taking the cube root of 125 and then squaring the result. This interpretation is key to performing the calculations accurately and efficiently.

Step 3: Evaluating the Cube Roots

The next critical step in solving $(0.064)^{\frac{2}{3}}$ is to evaluate the cube roots of the numerator and the denominator separately. We have $\frac{8^{\frac{2}{3}}}{125^{\frac{2}{3}}}$, which requires us to find the cube root of 8 and the cube root of 125. The cube root of a number is the value that, when multiplied by itself three times, equals the original number. In mathematical notation, the cube root of x is denoted as $\sqrt[3]{x}$.

The cube root of 8 is 2, because $2 \times 2 \times 2 = 8$. This is a fundamental cube root that is often encountered in mathematical problems. Similarly, the cube root of 125 is 5, because $5 \times 5 \times 5 = 125$. These two cube roots are essential building blocks for solving our expression.

Substituting these values back into our expression, we have $\frac{8^{\frac{2}{3}}}{125^{\frac{2}{3}}} = \frac{(\sqrt[3]{8})^2}{(\sqrt[3]{125})^2} = \frac{2^2}{5^2}$. This simplification significantly reduces the complexity of the problem, making it easier to arrive at the final answer. By finding the cube roots, we have transformed fractional exponents into simple integer powers, which are much easier to calculate.

The ability to recognize and evaluate common cube roots is a valuable skill in mathematics. It not only speeds up calculations but also enhances one's understanding of numerical relationships. In this case, knowing that the cube roots of 8 and 125 are 2 and 5, respectively, allows us to proceed smoothly to the next step in the problem-solving process. This step-by-step approach, breaking down a complex problem into smaller, more manageable parts, is a key strategy for success in mathematics.

Step 4: Squaring the Results

Having determined the cube roots of 8 and 125, we now have $\frac{2^2}{5^2}$. The next step is to square the results, which means raising each number to the power of 2. Squaring a number is simply multiplying it by itself. Thus, $2^2$ is $2 \times 2$, and $5^2$ is $5 \times 5$.

Calculating these squares, we find that $2^2 = 4$ and $5^2 = 25$. Substituting these values back into our expression, we get $\frac{4}{25}$. This fraction represents the final result of our calculations. We have successfully simplified the expression $(0.064)^{\frac{2}{3}}$ to its fractional form.

This step highlights the importance of understanding the order of operations in mathematics. We first dealt with the cube roots, as indicated by the denominator of the fractional exponent, and then we squared the results, as indicated by the numerator. This sequence ensures that we arrive at the correct answer. Squaring the results is a straightforward arithmetic operation, but it is a crucial step in completing the evaluation of the expression. It transforms the problem into its simplest form, allowing us to express the answer as a clear and concise fraction.

Step 5: Expressing the Answer

We have arrived at the fraction $\frac{4}{25}$ as the simplified form of $(0.064)^{\frac{2}{3}}$. This fraction represents the exact value of the expression. However, depending on the context, it may be desirable to express the answer as a decimal. To convert the fraction $\frac{4}{25}$ to a decimal, we divide the numerator (4) by the denominator (25).

Performing the division, we find that $\frac{4}{25} = 0.16$. This decimal form provides an alternative representation of the answer, which may be more convenient in some situations. For example, in practical applications, decimal values are often preferred for their ease of interpretation and use in calculations.

Thus, we can express the final answer in two equivalent forms: the fraction $\frac{4}{25}$ and the decimal 0.16. Both forms are correct and represent the same value. The choice of which form to use depends on the specific requirements of the problem or the preferences of the person using the result.

This final step underscores the flexibility of mathematical representation. We can express the same value in different forms, each with its own advantages and disadvantages. Being able to convert between fractions and decimals is a valuable skill that enhances one's mathematical fluency. In this case, we have shown how to convert a fraction to a decimal, completing the evaluation of the expression $(0.064)^{\frac{2}{3}}$ and providing a comprehensive answer.

Conclusion

In conclusion, the evaluation of $(0.064)^{\frac{2}{3}}$ demonstrates the power and elegance of fractional exponents. By converting the decimal to a fraction, applying the fractional exponent as a combination of a root and a power, evaluating the cube roots, squaring the results, and expressing the answer in both fractional and decimal forms, we have meticulously解剖d the problem and arrived at the solution. The final answer is $\frac{4}{25}$ or 0.16.

This exercise highlights several key mathematical concepts, including the conversion of decimals to fractions, the properties of exponents and roots, and the simplification of expressions. Mastering these concepts is crucial for success in mathematics and related fields. The step-by-step approach we have employed serves as a model for tackling similar mathematical challenges. By breaking down complex problems into smaller, more manageable parts, we can systematically work towards a solution.

The ability to work with fractional exponents is not just an academic skill; it has practical applications in various fields, including physics, engineering, and finance. Fractional exponents are used to model growth and decay, calculate compound interest, and describe relationships between physical quantities. Therefore, a thorough understanding of fractional exponents is a valuable asset in both academic and professional pursuits.

This comprehensive guide has provided a detailed explanation of how to evaluate $(0.064)^{\frac{2}{3}}$. By following the steps outlined in this article, readers can gain a deeper understanding of fractional exponents and develop the skills necessary to solve similar problems. The journey from the initial expression to the final answer showcases the beauty and precision of mathematics, where each step builds upon the previous one to reveal the underlying truth.