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Rules for Adding Variables with Exponents

Exponents in algebra represent repeated multiplication. In algebra, a variable is a letter that represents a value. Variables with exponents are read and pronounced as "variable raised to the power". For example, x^5 is pronounced as "x raised to the 5th power" or "x to the 5th". So, for example if a^4 means a times a times a times a, then if a = 4 then a^4 would be 4 times 4 times 4 times 4 = 256. Adding, subtracting, multiplying and dividing algebraic expressions each have different rules that involve first identifying like and unlike variables and exponents and then following the appropriate rules.
  1. Like Variables-Like Exponents

    • Adding single variable terms with exponents is a common operation in evaluating algebraic expressions. For example, 3x^2 + 5x^2 = (3+5) x^2 = 8x^2. The numbers 3 and 5 are called coefficients and should be added while the variable and exponents are kept the same. If the coefficient is not displayed in front of the variable, then assume it to be 1. For example: x^5 + x^5 = (1+1) x ^5 = 2x^5.

    Multiple Like Variables-Like Exponents

    • Adding multiple like variable terms where the exponents are also the same involves adding the coefficients and exponents while keeping the variables the same. An example: (1x^2 + 1y^2) + (1x^2 + 1y^2) would result in (1+1) x^2 + (1+1) y^2 = 2x^2 + 2y^2.

    Like Variables-Unlike Exponents

    • Adding multiple variable terms where exponents are unlike cannot be done. For example, 1x^2 + 1x^3 cannot be added because exponents are not the same. In this case, the expression is left in its simplest form.

    Unlike Variables-Like Exponents

    • Adding multiple variable terms with exponents where the exponents are both unlike cannot be done because the variables are not the same. For example, 1x^2 + 1y^2 cannot be added so must be left in its already simplest form. In addition, cases where the variables and exponents are unlike also cannot be added. So, the example, 2x^2 + 2y^3 cannot be added because the variables and exponents are not the same.

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