Evaluating The Expression [((3/5)^0)]^-2 Step-by-Step Solution

In mathematics, evaluating expressions is a fundamental skill. These expressions often involve various operations, exponents, and fractions. This article delves into the process of finding the value of a specific expression: [((3/5)^0)]-2. We will explore the underlying mathematical principles, step-by-step calculations, and the significance of zero and negative exponents. Understanding these concepts is crucial for simplifying complex expressions and solving mathematical problems.

Understanding the Expression

To accurately determine the value of the expression [((3/5)^0)]-2, we need to break it down into manageable parts and apply the relevant mathematical rules. The expression involves a fraction (3/5), a zero exponent, and a negative exponent. Let's first address the concept of exponents. An exponent indicates how many times a base number is multiplied by itself. For instance, in the expression 2^3, the base is 2, and the exponent is 3, which means 2 is multiplied by itself three times (2 * 2 * 2 = 8). The presence of a zero exponent and a negative exponent in our expression requires a specific understanding of these concepts. Any non-zero number raised to the power of 0 is defined as 1. This is a fundamental rule in mathematics that simplifies many expressions. A negative exponent indicates the reciprocal of the base raised to the positive exponent. For example, x^-n is equivalent to 1/(x^n). These rules are essential for simplifying and evaluating expressions with exponents. In the following sections, we will apply these rules step-by-step to find the value of the given expression.

Step-by-Step Calculation

To evaluate the expression [((3/5)^0)]-2, we will proceed step-by-step, applying the rules of exponents. First, we address the inner expression: (3/5)^0. According to the rule that any non-zero number raised to the power of 0 equals 1, we have:

(3/5)^0 = 1

Now, we substitute this result back into the original expression:

[1]^-2

Next, we deal with the negative exponent. A negative exponent indicates the reciprocal of the base raised to the positive exponent. In this case, 1^-2 means the reciprocal of 1^2:

1^-2 = 1/(1^2)

Since 1 raised to any power is 1 (1^2 = 1 * 1 = 1), we have:

1/(1^2) = 1/1 = 1

Therefore, the value of the expression [((3/5)^0)]-2 is 1. This step-by-step calculation demonstrates how the rules of zero and negative exponents are applied to simplify and evaluate mathematical expressions. Understanding these rules is crucial for handling more complex problems in algebra and calculus.

The Significance of Zero Exponents

The zero exponent rule is a fundamental concept in mathematics, stating that any non-zero number raised to the power of 0 is equal to 1. This might seem counterintuitive at first, but it is a necessary convention that maintains consistency in mathematical operations and rules. To understand why this rule exists, consider the pattern of exponents. For example, let's look at powers of 2:

2^3 = 8

2^2 = 4

2^1 = 2

Notice that as the exponent decreases by 1, the result is divided by 2. Following this pattern, if we decrease the exponent from 1 to 0, we would divide 2 by 2, which gives us 1. This leads us to:

2^0 = 1

This pattern holds true for any non-zero number. Mathematically, this can be expressed as:

x^0 = 1 (for x ≠ 0)

The zero exponent rule is crucial for simplifying algebraic expressions and equations. It ensures that mathematical operations remain consistent and predictable. In our original expression, (3/5)^0 simplifies to 1, which is a critical step in solving the problem. This rule is not just a mathematical quirk; it is a foundational principle that supports the structure of algebra and beyond.

The Role of Negative Exponents

Negative exponents are another essential concept in mathematics that extends the idea of exponents beyond positive integers. A negative exponent indicates the reciprocal of the base raised to the positive exponent. In other words, x^-n is equivalent to 1/(x^n). This concept is crucial for simplifying expressions and solving equations, particularly in algebra and calculus. To illustrate this, let's consider an example. Suppose we have 2^-3. According to the rule of negative exponents, this is equivalent to:

2^-3 = 1/(2^3) = 1/8

This shows how a negative exponent transforms the base into its reciprocal. The rule of negative exponents is particularly useful when dealing with fractions and reciprocals in algebraic expressions. It allows us to rewrite expressions in a more manageable form, making it easier to perform calculations and simplifications. In our original expression, [((3/5)^0)]-2, we encountered the term 1^-2. Applying the rule of negative exponents, we get:

1^-2 = 1/(1^2) = 1/1 = 1

This demonstrates how negative exponents play a critical role in simplifying and evaluating expressions. Understanding and applying the rule of negative exponents is essential for mastering algebraic manipulations and problem-solving in mathematics.

Common Mistakes to Avoid

When working with exponents, particularly zero and negative exponents, there are several common mistakes that students often make. Recognizing and avoiding these pitfalls is crucial for accurate calculations. One common mistake is assuming that any number raised to the power of 0 is 0. This is incorrect; any non-zero number raised to the power of 0 is 1. For example, 5^0 = 1, not 0. Another frequent error involves negative exponents. Students may incorrectly interpret a negative exponent as making the base negative. For instance, they might think that 2^-3 is equal to -8, which is wrong. The correct interpretation is that a negative exponent indicates the reciprocal of the base raised to the positive exponent. So, 2^-3 = 1/(2^3) = 1/8. Another mistake is mishandling the order of operations when dealing with exponents and other mathematical operations. It is essential to follow the correct order of operations (PEMDAS/BODMAS), which dictates that exponents should be evaluated before multiplication, division, addition, and subtraction. In the context of our original expression, [((3/5)^0)]-2, one might incorrectly apply the negative exponent before simplifying (3/5)^0. To avoid such mistakes, it is helpful to break down the expression step-by-step, applying the rules of exponents in the correct order. By being aware of these common mistakes and practicing careful calculation, you can improve your accuracy and confidence in working with exponents.

Conclusion

In conclusion, we have successfully found the value of the expression [((3/5)^0)]-2, which equals 1. This process involved understanding and applying the rules of zero and negative exponents. The zero exponent rule states that any non-zero number raised to the power of 0 is 1, while a negative exponent indicates the reciprocal of the base raised to the positive exponent. These concepts are fundamental in mathematics and are essential for simplifying expressions and solving equations. We also discussed the significance of these rules and the common mistakes to avoid when working with exponents. By breaking down the expression step-by-step and applying the rules correctly, we can confidently evaluate complex mathematical expressions. Mastering these concepts is crucial for further studies in algebra, calculus, and other advanced mathematical topics. The ability to manipulate and simplify expressions with exponents is a key skill in mathematical problem-solving. Understanding the underlying principles and practicing regularly will enhance your mathematical proficiency and accuracy.