Simplifying Exponential Expressions Step-by-Step Solution

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In the realm of mathematics, exponential expressions form a cornerstone of various disciplines, from basic algebra to advanced calculus and physics. Understanding how to manipulate and simplify these expressions is crucial for problem-solving and mathematical proficiency. In this article, we delve into the intricacies of simplifying a specific exponential expression, 2101â‹…21032206{\frac{2^{101} \cdot 2^{103}}{2^{206}}}, using the fundamental properties of exponents. Our goal is not just to arrive at the correct answer but to elucidate the underlying principles that govern exponential operations. We will embark on a step-by-step journey, unraveling the complexities of the expression and making it accessible to learners of all levels. This exploration will serve as a guide, empowering readers to tackle similar problems with confidence and precision. We will not only focus on the mathematical techniques but also emphasize the importance of understanding the logic behind each step. This holistic approach will foster a deeper appreciation for the elegance and power of exponential expressions in mathematics.

Understanding the Properties of Exponents

To effectively tackle the expression 2101â‹…21032206{\frac{2^{101} \cdot 2^{103}}{2^{206}}}, it is paramount to have a firm grasp of the properties of exponents. These properties serve as the foundational rules that dictate how exponents interact with each other during mathematical operations. Let's first outline these core principles, which will be our guiding lights throughout the simplification process.

  1. Product of Powers Property: When multiplying two exponential expressions with the same base, we add the exponents. Mathematically, this is represented as amâ‹…an=am+n{a^m \cdot a^n = a^{m+n}}. This property is essential for combining terms in the numerator of our expression.
  2. Quotient of Powers Property: When dividing two exponential expressions with the same base, we subtract the exponent in the denominator from the exponent in the numerator. This is expressed as aman=am−n{\frac{a^m}{a^n} = a^{m-n}}. This property will be crucial for simplifying the expression after we have applied the product of powers property.
  3. Power of a Power Property: While not directly applicable in this specific problem, it's worth noting that when raising a power to another power, we multiply the exponents: (am)n=amâ‹…n{(a^m)^n = a^{m \cdot n}}. This property is fundamental in more complex exponential simplifications.
  4. Zero Exponent Property: Any non-zero number raised to the power of 0 is equal to 1. That is, a0=1{a^0 = 1} if a≠0{a \neq 0}. This property is a special case that often appears during simplification.
  5. Negative Exponent Property: A negative exponent indicates the reciprocal of the base raised to the positive of that exponent. In other words, a−n=1an{a^{-n} = \frac{1}{a^n}}. This property is useful for converting expressions with negative exponents to their positive counterparts.

With these properties in mind, we are now well-equipped to approach the simplification of the given expression. Each property plays a unique role, and understanding their applications is key to mastering exponential expressions. In the subsequent sections, we will demonstrate how these properties are applied in a step-by-step manner to solve the problem at hand.

Step-by-Step Simplification of 2101â‹…21032206{\frac{2^{101} \cdot 2^{103}}{2^{206}}}

Now, let's embark on the simplification journey of the expression 2101â‹…21032206{\frac{2^{101} \cdot 2^{103}}{2^{206}}}. We will meticulously apply the properties of exponents discussed earlier to unravel the expression and arrive at its simplest form. This process will not only yield the solution but also provide a clear understanding of how exponential expressions can be manipulated.

Step 1: Applying the Product of Powers Property

The first step in simplifying the expression involves focusing on the numerator, which is 2101â‹…2103{2^{101} \cdot 2^{103}}. According to the product of powers property, when we multiply exponential expressions with the same base, we add the exponents. Therefore, we can rewrite the numerator as:

2101â‹…2103=2101+103=2204{2^{101} \cdot 2^{103} = 2^{101 + 103} = 2^{204}}

This step effectively condenses the two exponential terms in the numerator into a single term, making the expression more manageable. The application of this property is a cornerstone of simplifying exponential expressions, and it lays the groundwork for the subsequent steps.

Step 2: Applying the Quotient of Powers Property

With the numerator simplified to 2204{2^{204}}, our expression now looks like this:

22042206{\frac{2^{204}}{2^{206}}}

Next, we invoke the quotient of powers property, which states that when dividing exponential expressions with the same base, we subtract the exponent in the denominator from the exponent in the numerator. Applying this property, we get:

22042206=2204−206=2−2{\frac{2^{204}}{2^{206}} = 2^{204 - 206} = 2^{-2}}

This step further simplifies the expression by combining the numerator and the denominator into a single exponential term. The resulting exponent is negative, which leads us to the next property that we need to consider.

Step 3: Applying the Negative Exponent Property

We now have the expression 2−2{2^{-2}}. To eliminate the negative exponent, we utilize the negative exponent property, which tells us that a−n=1an{a^{-n} = \frac{1}{a^n}}. Applying this property, we can rewrite our expression as:

2−2=122{2^{-2} = \frac{1}{2^2}}

This step transforms the expression with a negative exponent into an equivalent expression with a positive exponent, making it easier to evaluate.

Step 4: Evaluating the Expression

Finally, we evaluate 22{2^2}, which is simply 2 multiplied by itself:

22=2â‹…2=4{2^2 = 2 \cdot 2 = 4}

Thus, our expression becomes:

122=14{\frac{1}{2^2} = \frac{1}{4}}

Converting this fraction to its decimal equivalent, we get:

14=0.25{\frac{1}{4} = 0.25}

Therefore, the simplified form of the expression 2101â‹…21032206{\frac{2^{101} \cdot 2^{103}}{2^{206}}} is 0.25. This step-by-step simplification not only provides the final answer but also illustrates the power and elegance of the properties of exponents in action.

Identifying the Correct Answer

Having meticulously simplified the expression 2101â‹…21032206{\frac{2^{101} \cdot 2^{103}}{2^{206}}} to 0.25, we now turn our attention to the multiple-choice options provided. This step is crucial in ensuring that our calculated result aligns with the given choices and that we select the correct answer.

The multiple-choice options presented are:

  • a) 4
  • b) 2
  • c) 1
  • d) 0.5
  • e) 0.25

By comparing our simplified result, 0.25, with the options, it becomes evident that option e) 0.25 is the correct answer. This step underscores the importance of not just performing the calculations accurately but also verifying the result against the provided choices.

Selecting the correct answer is the culmination of the simplification process. It validates our understanding of the properties of exponents and our ability to apply them effectively. In the subsequent sections, we will reinforce these concepts and provide additional insights to enhance your problem-solving skills in the realm of exponential expressions.

Common Mistakes to Avoid

Simplifying exponential expressions can be a straightforward process when the properties of exponents are applied correctly. However, certain common mistakes can lead to incorrect results. Being aware of these pitfalls is crucial for ensuring accuracy and developing a solid understanding of the subject matter. Let's explore some of the frequent errors that students and learners make when dealing with exponential expressions.

  1. Incorrectly Applying the Product of Powers Property: A common mistake is adding the bases instead of the exponents when multiplying exponential expressions with the same base. For example, mistaking 2101â‹…2103{2^{101} \cdot 2^{103}} for 4204{4^{204}} instead of 2204{2^{204}}. Remember, the product of powers property states that you should add the exponents, not the bases.

  2. Incorrectly Applying the Quotient of Powers Property: Similar to the product of powers property, a frequent error is subtracting the bases instead of the exponents when dividing exponential expressions with the same base. For instance, mistaking 22042206{\frac{2^{204}}{2^{206}}} for 0−2{0^{-2}} instead of 2−2{2^{-2}}. The quotient of powers property dictates that you should subtract the exponents, not the bases.

  3. Misunderstanding Negative Exponents: Negative exponents often cause confusion. A common mistake is interpreting a−n{a^{-n}} as −an{-a^n} instead of 1an{\frac{1}{a^n}}. For example, incorrectly thinking that 2−2{2^{-2}} is -4 instead of 14{\frac{1}{4}}. Remember that a negative exponent indicates the reciprocal of the base raised to the positive exponent.

  4. Forgetting the Order of Operations: When dealing with more complex expressions, it's essential to adhere to the order of operations (PEMDAS/BODMAS). Failing to do so can lead to incorrect simplifications. For example, if the expression involved additional operations, such as addition or multiplication, these should be addressed in the correct order.

  5. Ignoring the Zero Exponent Property: The zero exponent property, a0=1{a^0 = 1} (when a≠0{a \neq 0}), is sometimes overlooked. Any non-zero number raised to the power of 0 equals 1, and this should not be forgotten during simplification.

By being mindful of these common pitfalls and consistently applying the properties of exponents correctly, you can significantly reduce the chances of making errors and enhance your proficiency in simplifying exponential expressions.

Conclusion Mastering Exponential Expressions

In conclusion, the journey of simplifying the exponential expression 2101â‹…21032206{\frac{2^{101} \cdot 2^{103}}{2^{206}}} has provided a comprehensive overview of the fundamental properties of exponents and their practical applications. We embarked on a step-by-step simplification process, applying the product of powers, quotient of powers, and negative exponent properties to arrive at the solution, 0.25. This process not only yielded the correct answer but also illuminated the underlying principles that govern exponential operations. Furthermore, we identified and discussed common mistakes to avoid, equipping learners with the knowledge to tackle similar problems with confidence and accuracy.

Mastering exponential expressions is a crucial skill in mathematics, with applications spanning various fields, from algebra and calculus to physics and engineering. A solid understanding of the properties of exponents enables efficient problem-solving and fosters a deeper appreciation for the elegance and power of mathematical expressions. By adhering to the principles outlined in this guide, practicing consistently, and being mindful of potential pitfalls, you can enhance your proficiency in simplifying exponential expressions and unlock new dimensions of mathematical understanding. Remember, the key to success lies not just in memorizing the rules but in comprehending their logic and applying them judiciously.

e) 0.25