Expressions With A Value Of 16/81 A Detailed Solution

In the realm of mathematics, pinpointing expressions that evaluate to a specific value is a fundamental skill. In this comprehensive guide, we will delve into the intricacies of exponents and fractions to identify expressions that hold a value of 16/81. Our journey will involve dissecting each expression, applying the rules of exponents, and simplifying fractions to reveal the underlying value. Understanding how to manipulate these mathematical expressions is crucial for success in algebra, calculus, and beyond. Let's embark on this exploration to unlock the secrets hidden within these expressions and master the art of evaluating them. We'll break down each option step-by-step, providing clear explanations and insights along the way. Whether you're a student grappling with exponents for the first time or a seasoned math enthusiast looking to refresh your knowledge, this guide will equip you with the tools and understanding necessary to confidently tackle such problems. So, let's begin our mathematical adventure and unveil the expressions that proudly display a value of 16/81.

Evaluating (2/3)^4

To determine if the expression extbf{(2/3)^4} equals 16/81, we must apply the rules of exponents to both the numerator and the denominator. This expression signifies raising the fraction 2/3 to the fourth power, which translates to multiplying the fraction by itself four times. In mathematical terms, it can be written as (2/3) * (2/3) * (2/3) * (2/3). When we multiply fractions, we multiply the numerators together and the denominators together. Therefore, we have (2 * 2 * 2 * 2) / (3 * 3 * 3 * 3). Calculating the products, we find that 2 multiplied by itself four times (2^4) equals 16, and 3 multiplied by itself four times (3^4) equals 81. Consequently, the expression simplifies to 16/81. This outcome confirms that (2/3)^4 is indeed an expression with a value of 16/81. This methodical approach highlights the importance of understanding exponents and fraction manipulation in evaluating mathematical expressions. We have successfully demonstrated that this expression is a valid solution, and we will continue to analyze the remaining options using similar techniques.

Analyzing (16/3)^4

The next expression we need to evaluate is extbf{(16/3)^4}. Similar to the previous case, we apply the rules of exponents to both the numerator and the denominator. This means we raise both 16 and 3 to the fourth power. Mathematically, this can be represented as (16^4) / (3^4). Let's calculate each part separately. 16 raised to the fourth power (16^4) is 16 * 16 * 16 * 16, which equals 65,536. On the other hand, 3 raised to the fourth power (3^4) is 3 * 3 * 3 * 3, which we already know equals 81. Therefore, the expression simplifies to 65,536 / 81. Clearly, this fraction is significantly larger than 16/81. To further emphasize this difference, we can observe that the numerator, 65,536, is much greater than 16, while the denominator, 81, remains the same. This leads us to conclude that (16/3)^4 does not have a value of 16/81. This example underscores the critical role of careful calculation and comparison in determining the value of mathematical expressions. We have efficiently ruled out this option and can proceed to analyze the remaining expressions with greater focus.

Dissecting (4/81)^2

Now, let's turn our attention to the expression extbf{(4/81)^2}. Again, we apply the principle of raising both the numerator and the denominator to the power of 2. This means we square both 4 and 81. Mathematically, this translates to (4^2) / (81^2). Let's calculate these squares individually. 4 squared (4^2) is 4 * 4, which equals 16. 81 squared (81^2) is 81 * 81, which equals 6,561. Thus, the expression simplifies to 16/6,561. Comparing this fraction to our target value of 16/81, we notice that while the numerators are the same (16), the denominators are vastly different. 6,561 is significantly larger than 81. A larger denominator means the fraction represents a smaller value. Therefore, 16/6,561 is much smaller than 16/81. This observation allows us to confidently conclude that (4/81)^2 does not equal 16/81. This analysis demonstrates the importance of not only calculating but also comparing the resulting fractions to the target value. We have successfully eliminated another expression from our list and are narrowing down the possibilities.

Unraveling (4/9)^2

Next in our analysis is the expression extbf{(4/9)^2}. As we have done previously, we apply the rules of exponents by squaring both the numerator and the denominator. This can be represented as (4^2) / (9^2). Let's calculate these squares. 4 squared (4^2) is 4 * 4, which equals 16. 9 squared (9^2) is 9 * 9, which equals 81. Therefore, the expression simplifies to 16/81. This result matches our target value perfectly. Consequently, we can confidently state that (4/9)^2 is an expression with a value of 16/81. This successful identification reinforces the importance of meticulous calculation and the correct application of exponent rules. We have now found a second expression that satisfies the condition, adding to our understanding of the problem. The process of evaluating expressions and comparing them to a target value is a fundamental skill in mathematics, and this example further solidifies our grasp of this concept.

Investigating (1/81)^16

Finally, we examine the expression extbf{(1/81)^16}. This expression presents a slightly different challenge due to the larger exponent. However, the fundamental principles remain the same. We raise both the numerator and the denominator to the power of 16. This can be represented as (1^16) / (81^16). Let's analyze each part. 1 raised to any power is always 1. Therefore, 1^16 equals 1. Now, 81 raised to the power of 16 (81^16) is a very large number. We don't need to calculate the exact value to understand that it will be significantly larger than 81. As a result, the expression simplifies to 1 / (a very large number). This fraction will be an extremely small number, much smaller than 16/81. In fact, it will be very close to zero. Therefore, we can definitively conclude that (1/81)^16 does not equal 16/81. This analysis highlights the impact of large exponents on the value of fractions and the importance of understanding the relative magnitude of numbers. We have successfully analyzed all the given expressions and identified those that match our target value.

In conclusion, after meticulously evaluating each expression, we have identified that extbf{(2/3)^4} and extbf{(4/9)^2} are the only expressions with a value of 16/81. Our journey involved a step-by-step application of exponent rules and fraction simplification. We successfully demonstrated how to raise fractions to a power by raising both the numerator and the denominator to that power. We also emphasized the importance of comparing the resulting fractions to the target value to determine if they are equivalent. This exploration has reinforced the fundamental principles of exponents, fractions, and mathematical evaluation. The ability to confidently manipulate these expressions is crucial for success in various mathematical disciplines. By mastering these techniques, we empower ourselves to tackle more complex problems and deepen our understanding of the mathematical world. This guide has provided a comprehensive framework for evaluating such expressions, and we encourage continued practice to further solidify these skills.