Simplify Exponents: Integer & Rational Properties
Hey guys! Today, we're diving into the awesome world of exponents and how to simplify some pretty wild mathematical expressions. We're talking about using both integer and rational exponentiation properties to make things a whole lot easier. So, buckle up, because we're about to break down a complex problem step by step. Let's get started!
Understanding Exponent Properties
Before we jump into the main problem, let's quickly recap some essential exponent properties. These rules are the bread and butter of simplifying expressions, and knowing them inside out will make your life much easier. First up, we have the product of powers rule: when you multiply two powers with the same base, you add the exponents. Mathematically, it looks like this: a^m * a^n = a^(m+n). For example, 2^2 * 2^3 = 2^(2+3) = 2^5 = 32. Next, we've got the quotient of powers rule: when dividing two powers with the same base, you subtract the exponents: a^m / a^n = a^(m-n). So, 3^5 / 3^2 = 3^(5-2) = 3^3 = 27.
Then there's the power of a power rule: when you raise a power to another power, you multiply the exponents: (am)n = a^(mn). For instance, (52)3 = 5^(23) = 5^6 = 15625. Don't forget the power of a product rule: the power of a product is the product of the powers: (ab)^n = a^n * b^n. Like, (2*3)^2 = 2^2 * 3^2 = 4 * 9 = 36. Also, the power of a quotient rule: the power of a quotient is the quotient of the powers: (a/b)^n = a^n / b^n. For example, (4/2)^3 = 4^3 / 2^3 = 64 / 8 = 8. Lastly, anything raised to the power of zero is one (except zero itself): a^0 = 1 (where a ≠ 0). And remember that a negative exponent means you take the reciprocal: a^(-n) = 1/a^n. These properties are essential, so make sure you're comfortable with them before moving on! Understanding these rules thoroughly will help you tackle more complex expressions with confidence. Now that we've refreshed our knowledge, let's get back to the main event and simplify that expression!
Breaking Down the Problem
Okay, guys, let's tackle this beast of an expression: [(2n∙4(n+1))/(3∙8^n )∙9(2n)/16÷(3(2n) )2/12](-2). The key here is to break it down step by step, using our exponent rules to simplify each part. First, let's rewrite the expression using powers of prime numbers to make it easier to manage. We know that 4 = 2^2, 8 = 2^3, 9 = 3^2, and 16 = 2^4. Substituting these into our expression, we get: [(2n∙(22)(n+1))/(3∙(23)^n )∙(32)(2n)/24÷(3(2n) )2/12](-2). Now, let's simplify further using the power of a power rule: [(2n∙2(2n+2))/(3∙2^(3n) )∙3(4n)/24÷3(4n)/12](-2). See how much cleaner it's starting to look? By converting all numbers to their prime factor bases, we're setting ourselves up for easier simplification. Next, we'll combine the terms with the same base using the product and quotient rules. Remember, the goal is to consolidate and simplify each part of the expression before we deal with that outer exponent of -2.
Step-by-Step Simplification
Alright, let’s simplify the expression inside the brackets step by step. First, focus on the numerator: 2n∙2(2n+2) = 2^(n + 2n + 2) = 2^(3n+2). Then, consider the term 9^(2n) = (32)(2n) = 3^(4n). Now, rewrite the entire expression with these simplified terms: [(2(3n+2))/(3∙2(3n) )∙(3(4n))/24÷(3(4n))/12](-2). Next, let's tackle the division. Remember that dividing by a fraction is the same as multiplying by its reciprocal. So, we have: [(2(3n+2))/(3∙2(3n) )∙(3(4n))/24 * 12/(3(4n))](-2). Notice how we flipped the fraction and changed the division to multiplication. This is a crucial step in simplifying the expression. Now, we can rewrite 12 as 3 * 4, which is 3 * 2^2. So, our expression becomes: [(2(3n+2))/(3∙2(3n) )∙(3(4n))/24 * (3 * 22)/(3(4n))]^(-2). This might look a bit overwhelming, but we're getting closer! Next, we'll cancel out terms where possible and combine like terms to further simplify the expression.
Combining and Cancelling Terms
Okay, let's get to the fun part – cancelling and combining terms! We can rewrite the expression as: [(2^(3n+2) * 3^(4n) * 3 * 2^2) / (3 * 2^(3n) * 2^4 * 3(4n))](-2). Now, we can cancel out 3^(4n) from the numerator and denominator. This simplifies our expression to: [(2^(3n+2) * 3 * 2^2) / (3 * 2^(3n) * 24)](-2). Next, we can cancel out the factor of 3 from both the numerator and denominator: [(2^(3n+2) * 2^2) / (2^(3n) * 24)](-2). Now, let’s combine the powers of 2 in the numerator: 2^(3n+2) * 2^2 = 2^(3n+2+2) = 2^(3n+4). Similarly, in the denominator, we have: 2^(3n) * 2^4 = 2^(3n+4). So, our expression simplifies to: [2^(3n+4) / 2(3n+4)](-2). Wow, look at that! The numerator and denominator are now identical. This means the expression inside the brackets simplifies to 1: [1]^(-2). This is a massive simplification, and it shows the power of using exponent rules effectively.
Final Simplification
Alright, guys, we're in the home stretch! We've managed to simplify the entire expression inside the brackets down to just 1. So, now we have: [1]^(-2). Remember that any number raised to the power of zero is 1, and a negative exponent means taking the reciprocal. In this case, 1 raised to any power is still 1. So, 1^(-2) = 1/1^2 = 1/1 = 1. Therefore, the final simplified answer is 1. Isn't that satisfying? We took a complex expression with exponents and simplified it down to a single number. This example shows how crucial it is to understand and apply exponent properties correctly. Breaking down the problem into manageable steps, converting numbers to their prime factor bases, and carefully cancelling and combining terms are all key to simplifying these types of expressions. So, keep practicing, and you'll become a master of exponents in no time!
Conclusion
So, there you have it! We've successfully simplified the expression [(2n∙4(n+1))/(3∙8^n )∙9(2n)/16÷(3(2n) )2/12](-2) to 1 by applying the properties of integer and rational exponentiation. Remember, the key is to break down the problem into smaller, manageable steps and use the exponent rules to your advantage. Keep practicing, and you'll become more comfortable with these types of problems. Happy simplifying!