Simplifying √(12ⁿ * 15ⁿ⁺¹ * 20ⁿ⁺³): A Math Guide
Hey guys! Today, we're diving into a fun mathematical problem: simplifying the expression √(12ⁿ * 15ⁿ⁺¹ * 20ⁿ⁺³). This might look intimidating at first glance, but don't worry! We'll break it down step by step so it's super easy to understand. Get ready to put on your math hats and let's get started!
Understanding the Basics
Before we jump into the main problem, let's quickly recap some basic exponent rules and prime factorization. These concepts are crucial for simplifying complex expressions like the one we're tackling today. Trust me, a solid grasp of these basics will make the whole process a breeze! Believe me, you will need to understand this basic to solve the expression, so let's start.
Exponent Rules
Exponent rules are the foundation of simplifying expressions with powers. Here are a few key rules to keep in mind:
- Product of Powers: When multiplying terms with the same base, you add the exponents: aᵐ * aⁿ = aᵐ⁺ⁿ
- Power of a Power: When raising a power to another power, you multiply the exponents: (aᵐ)ⁿ = aᵐⁿ
- Power of a Product: The power of a product is the product of the powers: (ab)ⁿ = aⁿbⁿ
- Power of a Quotient: The power of a quotient is the quotient of the powers: (a/b)ⁿ = aⁿ/bⁿ
- Negative Exponent: A term raised to a negative exponent is the reciprocal of the term raised to the positive exponent: a⁻ⁿ = 1/aⁿ
- Zero Exponent: Any non-zero term raised to the power of zero is equal to 1: a⁰ = 1
Understanding and applying these rules will greatly simplify our task. They allow us to manipulate expressions, combine like terms, and reduce complexity. For example, knowing that aᵐ * aⁿ = aᵐ⁺ⁿ helps us combine terms when multiplying, while (aᵐ)ⁿ = aᵐⁿ allows us to simplify nested exponents. These rules are your best friends in the world of algebra, so make sure you're comfortable with them!
Prime Factorization
Prime factorization is the process of breaking down a number into a product of its prime factors. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself (e.g., 2, 3, 5, 7, 11, and so on). Factoring a number into its prime factors can simplify the original expression.
For example, let's prime factorize 12, 15, and 20:
- 12 = 2 × 2 × 3 = 2² × 3
- 15 = 3 × 5
- 20 = 2 × 2 × 5 = 2² × 5
Prime factorization is essential because it allows us to see the fundamental building blocks of a number. By expressing each number as a product of primes, we can easily identify common factors and simplify expressions. This technique is particularly useful when dealing with exponents and radicals, as it helps us break down complex terms into manageable components.
Breaking Down the Expression
Now that we've refreshed our memory on exponent rules and prime factorization, let's apply these concepts to our expression: √(12ⁿ * 15ⁿ⁺¹ * 20ⁿ⁺³). The first step is to express each term inside the square root in terms of its prime factors. This will help us simplify the expression and make it easier to work with.
Prime Factorize Each Term
Using the prime factorizations we found earlier:
- 12ⁿ = (2² × 3)ⁿ = 2²ⁿ × 3ⁿ
- 15ⁿ⁺¹ = (3 × 5)ⁿ⁺¹ = 3ⁿ⁺¹ × 5ⁿ⁺¹
- 20ⁿ⁺³ = (2² × 5)ⁿ⁺³ = 2²ⁿ⁺⁶ × 5ⁿ⁺³
Substitute Back into the Expression
Substitute these prime factorizations back into the original expression:
√(12ⁿ * 15ⁿ⁺¹ * 20ⁿ⁺³) = √(2²ⁿ × 3ⁿ × 3ⁿ⁺¹ × 5ⁿ⁺¹ × 2²ⁿ⁺⁶ × 5ⁿ⁺³)
Combine Like Terms
Now, let's combine the terms with the same base by adding their exponents:
√(2²ⁿ × 3ⁿ × 3ⁿ⁺¹ × 5ⁿ⁺¹ × 2²ⁿ⁺⁶ × 5ⁿ⁺³) = √(2^(2n + 2n + 6) × 3^(n + n + 1) × 5^(n + 1 + n + 3))
Simplify the exponents:
√(2^(4n + 6) × 3^(2n + 1) × 5^(2n + 4))
Simplifying the Square Root
Now that we have combined the like terms, we can simplify the square root. Remember that taking the square root of a term is the same as raising it to the power of ½. So, we need to divide each exponent by 2.
Apply the Square Root
√(2^(4n + 6) × 3^(2n + 1) × 5^(2n + 4)) = (2^(4n + 6) × 3^(2n + 1) × 5^(2n + 4))^½
Apply the power of ½ to each term:
= 2^((4n + 6)/2) × 3^((2n + 1)/2) × 5^((2n + 4)/2)
Simplify the exponents:
= 2^(2n + 3) × 3^(n + ½) × 5^(n + 2)
Rewrite the Expression
We can rewrite 3^(n + ½) as 3ⁿ × 3^(½) or 3ⁿ√3. This helps us separate the integer and fractional parts of the exponent.
= 2^(2n + 3) × 3ⁿ × √3 × 5^(n + 2)
Rearrange the Terms
Rearrange the terms to group the integer powers together:
= 2^(2n) × 2³ × 3ⁿ × 5ⁿ × 5² × √3
= (2²)^n × 8 × 3ⁿ × 5ⁿ × 25 × √3
= 4ⁿ × 8 × 3ⁿ × 5ⁿ × 25 × √3
Combine Like Terms Again
Combine 3ⁿ and 5ⁿ:
= 4ⁿ × (3 × 5)ⁿ × 8 × 25 × √3
= 4ⁿ × 15ⁿ × 200 × √3
Final Simplified Expression
So, the simplified expression is:
= 200 × (4 × 15)ⁿ × √3
= 200 × 60ⁿ × √3
Final Answer
Therefore, √(12ⁿ * 15ⁿ⁺¹ * 20ⁿ⁺³) simplifies to 200 * 60ⁿ * √3. Great job, guys! We made it through the simplification process. I hope you found this guide helpful and easy to follow. Math can be fun, and breaking down complex problems into smaller steps makes it much more manageable.
Key Takeaways
- Prime factorization is your friend. Always start by breaking down numbers into their prime factors.
- Exponent rules are essential. Make sure you know them inside and out.
- Take it step by step. Break down the problem into smaller, manageable steps.
- Don't be afraid to rewrite. Sometimes rewriting the expression in a different form can make it easier to simplify.
Keep practicing, and you'll become a math whiz in no time! And remember, if you ever get stuck, just break it down and take it one step at a time.