Which Expression Is Equivalent To 16^3 A Step-by-Step Solution

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Understanding exponents and their properties is crucial in mathematics. When faced with a problem like determining which expression is equivalent to 16316^3, it's essential to break down the components and apply the rules of exponents effectively. In this article, we will delve into the process of simplifying exponential expressions, specifically focusing on how to express 16316^3 in a different form using the base 2. We will explore the underlying principles, step-by-step solutions, and provide a detailed explanation to ensure a clear understanding. This guide aims to not only solve the given problem but also enhance your overall grasp of exponential manipulations.

Understanding the Basics of Exponents

Before diving into the specific problem, it’s important to understand the fundamental concepts of exponents. An exponent indicates how many times a base number is multiplied by itself. For example, in the expression ana^n, β€˜a’ is the base and β€˜n’ is the exponent. This means β€˜a’ is multiplied by itself β€˜n’ times. Grasping this concept is the first step in simplifying exponential expressions. The power of exponents is that they provide a concise way to represent repeated multiplication. Instead of writing 2imes2imes2imes22 imes 2 imes 2 imes 2, we can simply write 242^4. This notation is not only shorter but also makes it easier to perform calculations and manipulations, especially when dealing with large numbers or complex expressions. Understanding the components of an exponent – the base and the exponent itself – is crucial for successfully working with exponential expressions. This basic understanding forms the foundation for more complex operations such as simplifying expressions, solving equations, and understanding various mathematical concepts that rely on exponents.

Key Properties of Exponents

To effectively solve problems involving exponents, it's necessary to be familiar with some key properties. These properties allow us to simplify and manipulate exponential expressions. One fundamental property is the product of powers, which states that amimesan=am+na^m imes a^n = a^{m+n}. This means when multiplying two exponential terms with the same base, you can add their exponents. Another crucial property is the power of a power, represented as (am)n=amimesn(a^m)^n = a^{m imes n}. This indicates that when raising a power to another power, you multiply the exponents. There’s also the power of a product, which states that (ab)n=anbn(ab)^n = a^n b^n, showing that the power of a product is the product of the powers. Additionally, understanding the quotient of powers, am/an=amβˆ’na^m / a^n = a^{m-n}, is vital for simplifying division problems with exponents. Knowing that any number raised to the power of 0 equals 1, i.e., a0=1a^0 = 1, is another key aspect. Finally, negative exponents indicate reciprocals, such that aβˆ’n=1/ana^{-n} = 1/a^n. These properties are the building blocks for simplifying and solving exponential expressions and equations. By mastering these rules, you can efficiently manipulate and solve a wide variety of mathematical problems involving exponents.

Breaking Down the Problem: 16^3

Now, let's focus on the problem at hand: finding an expression equivalent to 16316^3. The first step is to recognize that 16 can be expressed as a power of 2. Specifically, 16=2416 = 2^4. This is a critical observation because it allows us to rewrite the expression 16316^3 in terms of the base 2, which is the base used in the answer options. By expressing 16 as a power of 2, we create a pathway to simplify the given expression and match it with one of the provided choices. This technique of converting numbers to a common base is a fundamental strategy in simplifying exponential expressions and solving related problems. Recognizing these relationships and applying them correctly is key to mastering exponential calculations. The ability to break down numbers into their prime factors and express them as powers is a valuable skill in mathematics, especially when dealing with exponents and logarithms.

Expressing 16 as a Power of 2

To express 16 as a power of 2, we need to determine how many times 2 must be multiplied by itself to equal 16. We can start by listing the powers of 2: 21=22^1 = 2, 22=42^2 = 4, 23=82^3 = 8, and 24=162^4 = 16. Thus, we find that 16=2416 = 2^4. This conversion is crucial because it allows us to rewrite the original expression, 16316^3, in terms of the base 2, which is essential for comparing it with the answer options provided. This step demonstrates the importance of recognizing numerical relationships and applying them to simplify mathematical expressions. The process of converting a number to a different base is a common technique in mathematics and computer science. Understanding how to do this efficiently can significantly simplify calculations and problem-solving. In this case, recognizing that 16 is a power of 2 is the key to unlocking the solution.

Applying the Power of a Power Rule

Once we've expressed 16 as 242^4, we can substitute this into the original expression, 16316^3. This gives us (24)3(2^4)^3. Now, we can apply the power of a power rule, which states that (am)n=amimesn(a^m)^n = a^{m imes n}. In our case, a=2a = 2, m=4m = 4, and n=3n = 3. Applying this rule, we get (24)3=24imes3=212(2^4)^3 = 2^{4 imes 3} = 2^{12}. This simplification is a direct application of one of the fundamental properties of exponents, illustrating how these rules can be used to efficiently manipulate and simplify expressions. The power of a power rule is particularly useful when dealing with nested exponents, as it allows us to combine them into a single exponent. This step is a clear demonstration of how understanding and applying the correct exponent rules can lead to a straightforward solution.

Step-by-Step Simplification

To reiterate, let's go through the step-by-step simplification:

  1. Recognize that 16=2416 = 2^4.
  2. Substitute 242^4 for 16 in the original expression: 163=(24)316^3 = (2^4)^3.
  3. Apply the power of a power rule: (24)3=24imes3(2^4)^3 = 2^{4 imes 3}.
  4. Multiply the exponents: 24imes3=2122^{4 imes 3} = 2^{12}.

This methodical approach ensures that each step is clear and logically sound, reducing the chance of errors. By breaking down the problem into smaller, manageable steps, we can confidently arrive at the correct solution. This step-by-step process is a valuable strategy for tackling mathematical problems, as it promotes clarity and accuracy. Each step builds upon the previous one, leading to a final simplified expression that is much easier to understand and compare with the answer options.

Identifying the Equivalent Expression

After simplifying 16316^3 to 2122^{12}, we can now easily identify the correct answer from the given options. The options were:

A. 272^7 B. 2112^{11} C. 2122^{12} D. 2642^{64}

Comparing our simplified expression, 2122^{12}, with the options, it is clear that option C, 2122^{12}, is the equivalent expression. This comparison highlights the importance of simplifying expressions to their simplest form to facilitate easy identification of equivalencies. The ability to manipulate expressions and reduce them to a common form is a key skill in mathematics. In this case, by expressing both the original expression and the answer options in terms of the base 2, we made the comparison straightforward and unambiguous. This final step underscores the effectiveness of the methods used throughout the solution process.

Why Other Options are Incorrect

To further solidify our understanding, let's briefly examine why the other options are incorrect:

  • A. 272^7: This is incorrect because 7 is not the result of multiplying the exponents after converting 16 to 242^4 and applying the power of a power rule.
  • B. 2112^{11}: This is also incorrect for the same reason as option A; 11 is not the correct exponent after simplification.
  • D. 2642^{64}: This is incorrect because it likely results from mistakenly cubing the exponent 4 (from 242^4) instead of multiplying it by 3.

Understanding why incorrect options are wrong can be as valuable as understanding why the correct option is right. It helps to reinforce the concepts and prevent similar mistakes in the future. By analyzing the potential errors that lead to these incorrect answers, we gain a deeper insight into the application of exponent rules and the importance of careful calculation. This critical analysis enhances our problem-solving skills and improves our overall mathematical understanding.

Conclusion

In conclusion, the expression equivalent to 16316^3 is 2122^{12}. We arrived at this answer by first recognizing that 16 can be expressed as 242^4, then applying the power of a power rule to simplify (24)3(2^4)^3 to 2122^{12}. This problem demonstrates the importance of understanding and applying the properties of exponents, particularly the power of a power rule, in simplifying expressions. By breaking down the problem into manageable steps and utilizing key exponent rules, we can efficiently solve complex mathematical problems. This approach not only provides the correct answer but also enhances our understanding of the underlying mathematical principles. Mastering these techniques is crucial for success in algebra and other advanced mathematical topics. The ability to manipulate exponents and simplify expressions is a fundamental skill that will be invaluable in various mathematical contexts.