Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of exponents and unraveling the mystery behind simplifying expressions. Our mission, should we choose to accept it, is to expand and simplify the expression (2² × (1/3²))⁻² and express the final answer as a positive power. Buckle up, because we're about to embark on a mathematical adventure!
Demystifying the Expression: (2² × (1/3²))⁻²
Before we jump into the expansion and simplification process, let's take a moment to understand what this expression actually represents. At its heart, it's a combination of exponents, fractions, and the crucial concept of a negative exponent. Remember, exponents tell us how many times a base number is multiplied by itself. For instance, 2² means 2 multiplied by itself, which equals 4. On the other hand, a fraction like 1/3² signifies a reciprocal, and the negative exponent throws in an extra twist, indicating that we need to take the reciprocal of the entire expression inside the parentheses.
The expression (2² × (1/3²))⁻² can be viewed as a challenge, an invitation to flex our mathematical muscles and demonstrate our understanding of exponent rules. It's a chance to transform something that might initially appear complex into a simplified, elegant form. So, let's roll up our sleeves and tackle this mathematical puzzle head-on!
Cracking the Code: The Step-by-Step Expansion
Now, let's get down to the nitty-gritty of expanding this expression. We'll break it down step by step, making sure each move is crystal clear. Our guiding principle will be the fundamental rules of exponents, which are like the secret code to unlocking these types of problems. Remember, a negative exponent means we need to take the reciprocal, and when we raise a product to a power, we distribute the power to each factor.
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Tackling the Negative Exponent: The first thing we notice is the negative exponent, -2. This tells us to take the reciprocal of the entire expression inside the parentheses. So, we can rewrite (2² × (1/3²))⁻² as 1 / (2² × (1/3²))². This simple step is crucial because it transforms the negative exponent into a positive one, making the rest of the calculation much smoother.
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Distributing the Power: Next, we have a power raised to another power. This is where the power of a product rule comes into play. We need to distribute the exponent 2 to both factors inside the parentheses: 2² and 1/3². This means we have 1 / (2²² × (1/3²)²). Remember, when we raise a power to another power, we multiply the exponents. So, 2²² becomes 2^(22) = 2⁴, and (1/3²)² becomes 1² / (3²)² = 1/3^(22) = 1/3⁴.
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Simplifying the Powers: Now, let's simplify the powers we've obtained. We know that 2⁴ means 2 multiplied by itself four times, which equals 16. Similarly, 3⁴ means 3 multiplied by itself four times, which equals 81. Substituting these values, our expression now looks like 1 / (16 × (1/81)). We're getting closer to the final answer!
Streamlining the Expression: The Art of Simplification
With the expansion complete, it's time to focus on simplification. Our goal is to tidy up the expression, combining terms and expressing the final result in its most concise form. This is where our understanding of fractions and basic arithmetic will shine.
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Multiplying Fractions: We have a product of 16 and 1/81. To multiply a whole number by a fraction, we can treat the whole number as a fraction with a denominator of 1. So, 16 becomes 16/1. Multiplying 16/1 by 1/81 gives us 16/81. Our expression now reads 1 / (16/81).
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Dividing by a Fraction: We're faced with dividing 1 by a fraction, 16/81. Remember, dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of 16/81 is 81/16. Therefore, 1 / (16/81) is equivalent to 1 × (81/16), which simply equals 81/16. We've successfully transformed the complex expression into a single fraction!
The Grand Finale: Expressing the Answer as a Positive Power
We're almost at the finish line! We've expanded the expression, simplified it, and now we need to express the answer as a positive power. Currently, our answer is in the form of a fraction, 81/16. To express this as a positive power, we need to recognize that both 81 and 16 are powers of other numbers.
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Recognizing the Powers: We know that 81 is 3⁴ (3 multiplied by itself four times) and 16 is 2⁴ (2 multiplied by itself four times). So, we can rewrite 81/16 as 3⁴ / 2⁴.
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Applying the Quotient Rule: We have a quotient with both the numerator and denominator raised to the same power. This is where the quotient rule of exponents comes into play. It states that a^n / b^n = (a/b)^n. Applying this rule, we can rewrite 3⁴ / 2⁴ as (3/2)⁴. And there you have it! We've successfully expressed the answer as a positive power.
The Triumphant Result: (3/2)⁴
After our mathematical journey, we've arrived at the final answer: (3/2)⁴. We started with a seemingly complex expression, (2² × (1/3²))⁻², and through careful expansion, simplification, and the application of exponent rules, we've transformed it into a concise and elegant form. This result not only showcases our understanding of exponents but also highlights the power of breaking down complex problems into manageable steps.
So, congratulations, fellow math adventurers! We've conquered this exponential challenge together. Remember, the world of mathematics is full of such exciting puzzles, waiting to be solved. Keep exploring, keep learning, and keep flexing those mathematical muscles!
In Summary:
- We began with the expression (2² × (1/3²))⁻².
- We tackled the negative exponent by taking the reciprocal.
- We distributed the power and simplified the exponents.
- We multiplied fractions and divided by a fraction.
- Finally, we expressed the answer as a positive power, arriving at (3/2)⁴.
Key Takeaways
- Negative Exponents: A negative exponent indicates the reciprocal of the base raised to the positive exponent. For example, x⁻ⁿ = 1/xⁿ.
- Power of a Product Rule: When raising a product to a power, distribute the power to each factor. For example, (ab)ⁿ = aⁿbⁿ.
- Power of a Power Rule: When raising a power to another power, multiply the exponents. For example, (aᵐ)ⁿ = a^(m*n).
- Quotient Rule: When dividing powers with the same exponent, divide the bases and raise the result to the exponent. For example, aⁿ / bⁿ = (a/b)ⁿ.
- Simplifying Fractions: Remember to simplify fractions by finding common factors and reducing them to their simplest form.
By mastering these concepts and practicing regularly, you'll become a true exponent expert! Keep up the great work, guys!