Exponents

Properties of Exponents

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

1. In the quantity 3x ⁵ , 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 r ² , 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 2x ² +5x ² . 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 2x ² +5x ² = (2 + 5)x ² = 7x ² .

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 = x ^7-8 = 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 = 3⁶ x ⁶ , and 3(4xy)⁵ = 3(4⁵)x ⁵ y ⁵ .
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≠a ² + b ² .

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 = 2^2×3 = 2⁶ and (x ⁵)^y = x ^5×y = x ^5y .

Writing Quantities with a Common Base

We cannot multiply quantities if they do not have the same base. However, we can often use the "power of a power" rule to write two quantities as quantities with a common base. If the two bases are powers of a common base (for example, 4 and 32 are both powers of 2 ), rewrite each base as a power of the common base. Then uses the power of a power rule to combine the exponents of each base. Finally, use the multiplication rule and add the exponents.

Example: 9⁵×27⁴ = ?

9⁵ = (3²)⁵ = 3¹0 ; 27⁴ = (3³)⁴ = 3¹2
9⁵×27⁴ = 3¹0×3¹2 = 3¹⁰⁺¹² = 3²2 .

Summary of Properties

Addition: cx ^a + dx ^a = (c + d )x ^a
Subtraction: cx ^a - dx ^a = (c - d )x ^a
Multiplication: cx ^a×dx ^b = (cd )x ^a+b
Division: = ()x ^a-b
Distribution: (cd )^a = c ^a d ^a
Power of a Power: c(x ^a)^b = cx ^ab