Product Rule

• The product rule of exponents states that if two identical base numbers with differing exponents are multiplied, the result is the base with the added exponents. For example, x^4 * x^5 = x^20. Practice this rule by working on increasingly complex examples. A harder example might be (-5(x^3)(y^4))(6(x^4)(y^2)), which becomes (-5 * 6) * (x^(3 + 4)) * (y^(4 + 2) = -30(x^7)(y^6).
  
  

Quotient Rule

• The quotient rule of exponents states that when like bases with differing exponents are presented as a fraction for division, the exponent of the denominator is subtracted from the exponent of the denominator. For example, (x^5) / (x^3) = x^2. Practice more difficult versions of this rule, such as problems including coefficients. For example, (6(x^3)(y^4)) / (12(x^2)(y^2)) = (6/12) * (x^ (3 - 2)) * (y^(4 - 2)) = (1/2)(x)(y^2) or (1/2)xy^2.
  
  

Negative Exponents

• If a base is raised by a negative exponent, its answer is represented by the inverse of that base raised to the absolute value, or positive version, of that exponent. For example, x^-2 is equal to 1 / (x^2). If the negative exponent presents in the denominator of a fraction, the result is a base to the positive exponent. For example, 1 / (x^-3) becomes x^3. Note that when numbers are used in place of variables, the expression can be simplified by performing the exponent operation. For example, 3^-3 becomes 1 / (3^3) or 1 / 27.
  
  

Power Rules

• The power rule for exponents states that if a base raised to an exponent is raised to another exponent, outside of the parentheses, then the two exponents should be multiplied. For example, (x^5)^3 becomes x^15.
  
  The product to power rule states that for two bases multiplied within parenthesis with an external exponent, the result is each base raised to that exponent. For example, (xy)^7 becomes x^7 * y^7. The quotient to power rule similarly states that an external exponent applies to both numerator and denominator of an interior quotient. For example, (2/5)^2 becomes (2^2) / (5^2) or (4/25).
  
  

Combining Rules

• Practice working with each rule individually until you are confident. Then begin working on problems that include multiple rules in one expression. For example, (3 / x)^-2 involves the power rule of quotients and a negative exponent. The inverse of the interior quotient can be represented by flipping the fraction and then applying the absolute value of the exponent to each portion: (3 / 5)^-2 becomes (5^2) / (3^2) or 25 / 9.