In addition to possessing the relevant algebraic and arithmetic background knowledge, you must also get acquainted with some general guidelines before attempting to perform operations with exponents. Any variable raised to the zero power simplifies to 1, and any variable without an exponent displayed is assumed to have an exponent of 1. Algebraically, these laws are represented by x^0 = 1 and x^1 = x. It is also essential to know that you cannot add, subtract, multiply or divide exponents whose bases differ from one another. For example, r^3 + q^4 and m^7/j^7 cannot be simplified further. And, when writing solutions that involve multiple variables, order doesn’t matter. For instance, (t^4)(s^3) and (s^3)(t^4) are equivalent.
If adding or subtracting exponents, add or subtract only the coefficients, leaving the exponents and variables unchanged. Consider the expression 5x^7 – 2x^7. To simplify this, subtract two from five, resulting in 3x^7. Keep in mind, however, that you cannot add or subtract terms whose exponents do not match. In such situations, terms cannot be combined; they must be left as-is. For instance, suppose that you are asked to simplify x^2 + x^3. Many people assume that the answer would be x^5. But the correct answer is x^2 + x^3.
When multiplying powers that have the same base, add the exponents. Consider -6y^4 * 2y^5. Multiply the coefficients, -6 and 2, to get -12, and then add the exponents, producing y^9. In total, this expression simplifies to -12y^9. If an expression contains multiple bases, multiply like bases by one another. For example, multiplying 5*f^3*n^2*f^4*n results in 5(f^7)(n^3).
When dividing powers that share the same base, subtract the exponent in the denominator from the exponent in the numerator. (See Reference 5) Consider 8x^9/4x^3. Divide the coefficients, 8 and 4, resulting in 2. Subtract the exponents, 9 and 3, resulting in x^6. This produces a solution of 2x^6. If the coefficient in the numerator is larger than the coefficient in the numerator, the answer will be a fraction or decimal. For instance, simplifying 18y^6/24y^2 yields 3y^4/4 or 0.75y^4.
Sometimes you might encounter a problem in which an exponent lies outside a set of parentheses. In these types of problems, simply multiply the exponents, and if there are any coefficients, raise them to the power of the exponent listed outside of the parentheses. For example, (3h^5)^2 simplifies to 9h^10, because the squaring the coefficient produces nine and multiplying the exponents equals 10.
As with coefficients, exponents can also be negative. You can make a negative exponent positive by turning it into a fraction. Place the coefficient in the numerator of the fraction and the variable and its exponent in the denominator. For instance, 5x^-9 becomes 5/x^9.