%5E2%2B2mn%3D16%2F9m2%2B9%2F16n2.jpeg "(4/3m-3/4n)^2+2mn=16/9m2+9/16n2")
Expanding and Simplifying the Equation: (4/3m - 3/4n)^2 + 2mn = 16/9m^2 + 9/16n^2
This equation presents a challenge in algebra, requiring us to expand the square and simplify the expression to see if it holds true. Let's break it down step by step:
Expanding the Square
First, we need to expand the square term on the left side of the equation:
(4/3m - 3/4n)^2 = (4/3m - 3/4n) * (4/3m - 3/4n)
Using the FOIL (First, Outer, Inner, Last) method, we get:
- First: (4/3m) * (4/3m) = 16/9m^2
- Outer: (4/3m) * (-3/4n) = -mn
- Inner: (-3/4n) * (4/3m) = -mn
- Last: (-3/4n) * (-3/4n) = 9/16n^2
Combining the terms, we get:
(4/3m - 3/4n)^2 = 16/9m^2 - 2mn + 9/16n^2
Substituting and Simplifying
Now we can substitute this expanded term back into the original equation:
16/9m^2 - 2mn + 9/16n^2 + 2mn = 16/9m^2 + 9/16n^2
Notice that the -2mn and +2mn terms cancel each other out. This leaves us with:
16/9m^2 + 9/16n^2 = 16/9m^2 + 9/16n^2
Conclusion
Therefore, we have proven that the original equation holds true. The expanded and simplified expression on both sides of the equation are identical.