Final answer:
To solve 12⁷⋅ 30⁵/25²⋅ 8⁵⋅ 27⁴, we prime factorize the bases and apply the laws of exponents. We add exponents for multiplication and subtract for division when bases are the same. After simplifying, we arrive at a final expression of 2⁴⋅ 3¹⁰ / 5⁴.
Step-by-step explanation:
To find the value of 12⁷⋅ 30⁵/25²⋅ 8⁵⋅ 27⁴, we apply the laws of exponents. First, each base raised to a power must be dealt with individually. For the division, we subtract exponents when the base is the same and for multiplication, we add exponents when the base is the same. The challenge here is that the bases are not the same, but we can prime factorize each to get to a common base if possible.
Let's break it down:
- Prime factorization of bases: 12 = 2²⋅ 3, 30 = 2⋅ 3⁵, 25 = 5², 8 = 2³, and 27 = 3³.
- Rewrite with prime factors: (2²⋅ 3)⁷ ⋅ (2⋅ 3⁵)⁵ / (5²)²⋅ (2³)⁵⋅ (3³)⁴.
- Expand using laws of exponents: 2¹⁴⋅ 3⁷ ⋅ 2⁵⋅ 3¹⁵ / 5⁴⋅ 2¹⁵⋅ 3¹².
- Simplify by adding and subtracting exponents where bases are the same: 2¹⁴+5-15⋅ 3⁷+25-12 / 5⁴.
- Simplify further: 2⁴⋅ 3¹⁰ / 5⁴.
- If necessary, compute the final value using a calculator or continue simplifying if common factors exist.
Remember that the power affects everything inside the parentheses, so you must apply the exponents to both the coefficients and the variables when expanding.