OBJECTIVE: The students will simplify expressions by using the laws

OBJECTIVE: The students will simplify expressions by using the laws of exponents.

Write in exponential form 3x3 2x2x2x2x2x2 10x10x10 5x5x5 1x1x1x1x1x1x1x1

HISTORY of the word EXPONENT. The term EXPONENT was introduced by Michael Stifel (14871567) in 1544 in Arithmetica integra.

An exponent is simply shorthand for multiplying that number of identical factors.

So 4³ is the same as (4)(4)(4), three identical factors of 4. And x³ is just three factors of x, (x)(x)(x).

Exponent is the power in an expression. 13 7

Exponents exponent 5 3 power base Example: 125 53 means that 53 is the exponential form of the number 125.

Using the Calculator 54 Press 5 Press Press 4 Then

To 7 Laws of Exponents #1 PRODUCT LAW Multiply LIKE Bases Copy the Base, Add Exponents

Product Law or Product Rule m n x x x 3 4 m n Example: 2 2 2 3 4 2 7 Proof: 23 2 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 7

7 Laws of Exponents #2 QUOTIENT LAW To Divide LIKE Bases Subtract 6 Exponents a 2 a a 4

Quotient Law or Quotient Rule: m x m n m n x x x n x 4 5 4 3 4 3 1 Example: 3 5 5 5 5 5 5 4 5 5 5 5 5 Proof: 3 5 5 5 5 5

7 Laws of Exponents #3 EXPONENT of EXPONENT LAW To Raise a Power to a Power Multiply Exponents 3 4 a a12

Exponent of Exponent Law or Exponential Rule: n m x Example: 4 Proof: 4 3 2 3 2 x 4 2 3 2 mn 4 6 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 6

7 Laws of Exponents #4 Raising a product to a power To Raise a QUANTITY to a Power, raise EACH Factor to that Power. 2 2 2 (-3ab) 9a b

7 Laws of Exponents #5 Raising a quotient or a fraction to a power To Raise a FRACTION to a Power, raise BOTH Numerator & Denominator to that power. 4 4 a a 4 b

x y n n x n y 3 3 2 2 Example: 3 7 7

7 Laws of Exponents #6 NEGATIVE EXPONENT LAW Negative Exponents Reciprocal with a Positive Exponent 3 a a 3

#6: Negative Law of Exponents: If the base is powered by the negative exponent, then the base becomes reciprocal with the positive exponent. x m 1 m x Example #1: 2 3 1 1 3 2 8 3 1 5 3 Example #2: 3 5 125 5 1

7 Laws of Exponents #7 Any ONE nonzero number raised to the ZERO Power 0 2 0 a 0 3, 263,546

The Laws of Exponents: #7: Zero Law of Exponents: Any base powered by zero exponent equals one 0 x 1 Example: 112 5 7 0 1 flower 0 1 0 1

3 Basic Examples 3 x x 3 y y 7 7 4 x x 3 x 4 x 1 5 x 1 1 7 7 5 2 x x x

Basic Examples 2 3 x x x 4 3 x xy 3 x 2 3 x 12 4 3 x 3 3 x y 5