Equivalent Expressions For (2^5)^-2 A Step By Step Solution

This mathematical puzzle delves into the realm of exponents, challenging us to decipher equivalent expressions for the given term (2⁵⁾-2. To unravel this, we'll embark on a step-by-step journey, dissecting the properties of exponents and exploring various transformations that lead us to the correct equivalents. This exploration will not only solidify our understanding of exponent rules but also enhance our problem-solving skills in mathematics.

Demystifying Exponents: A Foundation for Equivalence

At the heart of this mathematical exploration lies the concept of exponents. An exponent indicates the number of times a base is multiplied by itself. In the expression (2⁵⁾-2, 2 is the base, and 5 and -2 are exponents. To decipher the equivalent expressions, we must first understand the fundamental rules governing exponents. Key among these is the power of a power rule, which states that (a^m)n = a^(m*n). This rule is our primary tool in simplifying the given expression. Applying this rule, we transform (2⁵⁾-2 into 2^(5-2)*, which further simplifies to 2^-10. This crucial step unveils our first equivalent expression: 2^-10. But our quest doesn't end here. We need to explore other forms of this expression to identify all the equivalents.

The negative exponent rule comes into play next. This rule dictates that a^-n = 1/a^n. Applying this to 2^-10, we get 1/2^10. Now, we need to compute 2 raised to the power of 10. This is where our computational skills are put to the test. Multiplying 2 by itself ten times (222222222*2) yields 1024. Thus, 1/2^10 is equivalent to 1/1024. This unveils our second equivalent expression. We've successfully navigated the realm of exponents, transforming the initial expression into two equivalent forms: 2^-10 and 1/1024. This journey showcases the power of exponent rules in simplifying complex expressions and revealing their hidden equivalencies.

Exploring the Proposed Equivalents: A Critical Analysis

Now that we've established the equivalent expressions for (2⁵⁾-2 as 2^-10 and 1/1024, let's critically analyze the options provided to determine which ones match our findings. We have four options to evaluate:

1. 2^-10 and 1/20
2. 2^-10 and 1/1024
3. 10^-2 and 1/100
4. 10^-10

The first option presents 2^-10, which we've already identified as a correct equivalent. However, it pairs it with 1/20. To determine if 1/20 is indeed equivalent, we can compare it with our established equivalent, 1/1024. Clearly, 1/20 and 1/1024 are vastly different values, making the first option incorrect. The second option offers 2^-10 and 1/1024, both of which align perfectly with our derived equivalents. This option emerges as a strong contender. The third option proposes 10^-2 and 1/100. While 10^-2 is indeed equivalent to 1/100 (10^-2 = 1/10^2 = 1/100), these expressions are not equivalent to our original (2⁵⁾-2. This option is therefore incorrect. The fourth option simply states 10^-10. While it's a valid mathematical expression, it doesn't match our derived equivalents for (2⁵⁾-2. Thus, it's also incorrect. Through this meticulous analysis, we pinpoint the second option, 2^-10 and 1/1024, as the sole pair of expressions equivalent to the initial expression. This exercise underscores the importance of not just simplifying expressions but also critically evaluating proposed solutions against established truths.

The Correct Answer: 2^-10 and 1/1024

After a thorough exploration of exponent rules and a meticulous analysis of the provided options, we confidently arrive at the definitive answer: the expressions equivalent to (2⁵⁾-2 are 2^-10 and 1/1024. This conclusion is grounded in the application of the power of a power rule, which transformed (2⁵⁾-2 into 2^-10, and the negative exponent rule, which further converted 2^-10 into 1/2^10, ultimately yielding 1/1024. Our journey has not only provided the correct answer but also reinforced our understanding of exponent manipulation and the importance of step-by-step simplification. This rigorous process ensures accuracy and builds confidence in our mathematical abilities. By breaking down the problem into manageable steps and applying the relevant rules, we've successfully navigated the complexities of exponents and arrived at a clear and concise solution.

Why the Other Options are Incorrect: A Detailed Explanation

To further solidify our understanding, let's delve into why the other options presented are incorrect. This will not only reinforce the correct solution but also help us identify common errors and misconceptions in exponent manipulation.

• Option 1: 2^-10 and 1/20

  We've already established that 2^-10 is a correct equivalent. However, 1/20 is where the error lies. The confusion might stem from mistakenly applying the negative exponent rule directly to the base and exponent without considering the power of a power rule first. Remember, (2⁵⁾-2 simplifies to 2^-10, which means 1 divided by 2 raised to the power of 10, not 2 raised to the power of 1. This highlights the importance of following the correct order of operations and applying the appropriate rules sequentially.

• Option 3: 10^-2 and 1/100

  This option presents a correct equivalence in itself: 10^-2 is indeed equal to 1/100. However, these expressions are not equivalent to our original expression, (2⁵⁾-2. The base is different (10 versus 2), and the resulting value is significantly different. This underscores the crucial point that we're looking for expressions equivalent to the specific expression given, not just any mathematically valid equivalence. It is a reminder to be mindful of the base and the overall value when simplifying and comparing expressions.

• Option 4: 10^-10

  Similar to option 3, this option presents a mathematically valid expression but fails to be equivalent to (2⁵⁾-2. The base is different, and the value is vastly smaller. 10^-10 represents 1 divided by 10 raised to the power of 10, a minuscule number, while (2⁵⁾-2 simplifies to 1/1024, a significantly larger value. This reinforces the importance of maintaining equivalence throughout the simplification process and ensuring that the final expression represents the same value as the initial one.

By dissecting these incorrect options, we gain a deeper appreciation for the nuances of exponent manipulation and the importance of adhering to the rules meticulously. This analysis not only validates our correct answer but also equips us with the knowledge to avoid similar errors in the future.

Conclusion: Mastering Exponent Equivalence

In conclusion, the expressions equivalent to (2⁵⁾-2 are definitively 2^-10 and 1/1024. This determination stemmed from a systematic application of exponent rules, specifically the power of a power rule and the negative exponent rule. We meticulously simplified the initial expression, explored potential equivalents, and critically analyzed each option to arrive at the correct answer. Furthermore, we delved into the reasons why the other options were incorrect, highlighting common errors and misconceptions in exponent manipulation. This comprehensive exploration not only solved the problem at hand but also fortified our understanding of exponents and their properties. Mastering exponent equivalence is crucial for success in various mathematical domains, and this exercise serves as a valuable step in that journey. By practicing these concepts and diligently applying the rules, we can confidently tackle more complex mathematical challenges and unlock the power of exponents.