%5E3.jpeg "(4m^5n^2/6m^2n)^3")
Simplifying (4m^5n^2/6m^2n)^3
This problem involves simplifying a complex expression with exponents and fractions. Let's break it down step by step.
Understanding the Properties of Exponents
Before we begin, let's recall some important properties of exponents:
- Product of powers: x^m * x^n = x^(m+n)
- Quotient of powers: x^m / x^n = x^(m-n)
- Power of a power: (x^m)^n = x^(m*n)
- Power of a product: (x*y)^n = x^n * y^n
- Power of a quotient: (x/y)^n = x^n / y^n
Simplifying the Expression
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Simplify the fraction inside the parentheses: (4m^5n^2/6m^2n) = (2/3) * m^(5-2) * n^(2-1) = (2/3)m^3n
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Apply the power of a product rule: ((2/3)m^3n)^3 = (2/3)^3 * (m^3)^3 * (n)^3
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Simplify the exponents: (2/3)^3 * m^(3*3) * n^3 = 8/27 * m^9 * n^3
Final Result
Therefore, the simplified form of (4m^5n^2/6m^2n)^3 is (8/27)m^9n^3.