The Product Rule of Exponents states that when you multiply two expressions that have the same base, you add their exponents. That is:
\[ a^m \cdot a^n = a^{m+n} \]
To determine an expression that simplifies to 2412 using the Product Rule of Exponents, we first need to factor 2412 into its prime bases. Let's start by doing the prime factorization of 2412:
- Divide by 2: \[ 2412 \div 2 = 1206 \]
- Divide by 2 again: \[ 1206 \div 2 = 603 \]
- Divide 603 by 3: \[ 603 \div 3 = 201 \]
- Divide 201 by 3 again: \[ 201 \div 3 = 67 \]
- 67 is a prime number.
Now we can express 2412 as a product of its prime factors: \[ 2412 = 2^2 \cdot 3^2 \cdot 67^1 \]
If we want to express 2412 using the Product Rule of Exponents, we can take two bases and their respective powers, such as:
\[ (2^2 \cdot 3^2) \cdot 67^1 \]
Which would correctly use the Product Rule if established as: \[ 2^2 \cdot 3^2 \cdot 67^1 = 4 \cdot 9 \cdot 67 = 2412 \]
More simply, we could also write: \[ (2^2 \cdot 3^2) \cdot 67 = 4 \cdot 9 \cdot 67 \]
so you could multiply 2^2 and 3^2 together and then multiply that result by 67.
Thus, an expression that simplifies to 2412 using the Product Rule could be: \[ 2^2 \cdot 3^2 \cdot 67^1 \]
Or any rearrangement where the bases and their respective powers are multiplied together.
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