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Let's solve each of the equations step by step: 1. ( (a+b) = a+ b ) - This equation holds true for all values of ( a ) and ( b ) because both sides are equal. 2. ( (a-b)^2 = a^2 + b^2 – 2ab ) - This is a standard expansion of the square of a binomial. 3. ( (a+b)(a-b) = a^2 – b^2 ) - This is the difference of squares identity. 4. ( (x + a)(x + b) = x^2 + (a + b)x + ab ) - This is the expansion of the product of two binomials. 5. ( (x + a)(x – b) = x^2 + (a – b)x – ab ) - This is also the expansion of the product of two binomials. 6. ( (a + b)^3 = a^3 + b^3 + 3ab(a + b) ) - This is the expansion of the cube of a binomial. 7. ( (a-b)^3 = a^3 – b^3 – 3ab(a – b) ) - This is also the expansion of the cube of a binomial. 8. ( (x – a)(x + b) = x^2 + (b – a)x – ab ) - This is the expansion of the product of two binomials. 9. ( (x – a)(x – b) = x^2 – (a + b)x + ab ) - This is also the expansion of the product of two binomials. 10. ( (x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2xz ) - This is the expansion of the square of a trinomial. 11. ( (x + y – z)^2 = x^2 + y^2 + z^2 + 2xy – 2yz – 2xz ) - This is also the expansion of the square of a trinomial. 12. ( (x – y + z)^2 = x^2 + y^2 + z^2 – 2xy – 2yz + 2xz ) - This is also the expansion of the square of a trinomial. 13. ( (x – y – z)^2 = x^2 + y^2 + z^2 – 2xy + 2yz – 2xz ) - This is also the expansion of the square of a trinomial. 14. ( x^2 + y^2 + z^2 – 3xyz = (x + y + z)(x^2 + y^2 + z^2 – xy – yz -xz) ) - This is an identity that can be verified by expanding the right side. 15. ( y^2 + z^2 = frac{1}{2}[(x + y)^2 + (x – y)^2] ) - This is an identity that relates the squares of terms in terms of their sums and differences. 16. ( x^2 + y^2 + z^2 -xy - yz - zx = frac{1}{2}[(x-y)^2 + (y-z)^2 + (z-x)^2] ) - This is also an identity that relates the squares of terms in terms of their differences. These equations arIf you're reading this far, you don't have to solve this question just type anything lol, had to use chat GPTe mostly identities or expansions that are commonly used in algebraic manipulations. They're fundamental tools for solving equations and simplifying expressions in mathematics.