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A quantity with an exponent has three components--the base, the exponent, and the coefficient.

  1. In the quantity 3x5, the coefficient is 3, the base is x, and the exponent is 5.
  2. In the quantity 3(16)7x, the coefficient is 3, the base is 16, and the exponent is 7x.
  3. In the quantity 26(2y)xy, the coefficient is 26, the base is 2y, and the exponent is xy.
  4. In the quantity r2, the (implied) coefficient is 1, the base is r, and the exponent is 2.

Adding and Subtracting Quantities with Exponents

We cannot simplify by grouping two terms together unless they have the same base and the same exponent. For example, we cannot combine terms in expressions such as 52 +122 or 53 +54. We can, however, simplify 45 +45 and 2x2 +5x2. To group two terms with the same base and the same exponent, add their coefficients. Thus, 45 +45 = 1(4)5 + 1(4)5 = (1 + 1)(4)5 = 2(4)5 and 2x2 +5x2 = (2 + 5)x2 = 7x2.

Multiplying Quantities with Exponents

We can multiply two quantities with exponents if they have the same base. To multiply two quantities with the same base, multiply their coefficients and add their exponents. For example, 4(5)5×3(5)2 = (3×4)(5)5+2 = 12(5)7 and 5(2x)2×6(2x)y = (5×6)(2x)2+y = 30(2x)2+y.

Dividing Quantities with Exponents

We can divide two quantities with exponents if they have the same base. To divide two quantities with the same base, divide their coefficients and subtract their exponents. For example, = (2)11-6 = 3(2)5 and = x7-8 = x-1.

Distributive Property of Exponents

If an exponent acts on single term in parentheses, we can distribute the exponent over the term. For example, (2×5)2 = (22)(52), (3x)6 = 36x6, and 3(4xy)5 = 3(45)x5y5.
Be careful! If an exponent acts on multiple terms in parentheses (i.e. if there is a "+" or "-" sign in the parentheses), it cannot be distributed: (5 + 3)2≠52 +32 and (4a + b)2≠a2 + b2.

Taking a Power of a Power

Sometimes, the base will include an exponent, like in the expression (22)3. If this is the case, multiply the exponent in the base by the exponent which acts on the base: (22)3 = 22×3 = 26 and (x5)y = x5×y = x5y.

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