%5E2-x-(mn%5E4%2F9n)%5E2.jpeg "(3m^-5n^2/4m^-2n^0)^2 X (mn^4/9n)^2")
Simplifying the Expression: (3m^-5n^2/4m^-2n^0)^2 x (mn^4/9n)^2
This problem involves simplifying a complex expression with exponents and variables. Let's break it down step-by-step:
Understanding the Rules
Before we start simplifying, let's recall some key rules for 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)
- Anything raised to the power of zero equals 1: x^0 = 1
Simplifying the First Part
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Simplify the inside of the first parentheses: (3m^-5n^2 / 4m^-2n^0) = (3/4) * (m^-5/m^-2) * (n^2/n^0) Applying the quotient of powers rule, this becomes: (3/4) * m^(-5+2) * n^(2-0) = (3/4) * m^-3 * n^2
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Square the simplified expression: [(3/4) * m^-3 * n^2]^2 = (3/4)^2 * (m^-3)^2 * (n^2)^2 Applying the power of a power rule, we get: (9/16) * m^(-32) * n^(22) = (9/16) * m^-6 * n^4
Simplifying the Second Part
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Simplify the inside of the second parentheses: (mn^4 / 9n) = (1/9) * (m/1) * (n^4/n) Applying the quotient of powers rule, we get: (1/9) * m * n^(4-1) = (1/9) * m * n^3
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Square the simplified expression: [(1/9) * m * n^3]^2 = (1/9)^2 * m^2 * (n^3)^2 Applying the power of a power rule, we get: (1/81) * m^2 * n^(3*2) = (1/81) * m^2 * n^6
Combining the Simplified Parts
Now, we have: (9/16) * m^-6 * n^4 * (1/81) * m^2 * n^6
- Multiply the coefficients: (9/16) * (1/81) = 1/144
- Apply the product of powers rule for 'm': m^-6 * m^2 = m^(-6+2) = m^-4
- Apply the product of powers rule for 'n': n^4 * n^6 = n^(4+6) = n^10
Final Result
The simplified expression is: 1/144 * m^-4 * n^10
This can also be written as: n^10 / (144 * m^4)