P.2 Exponents and Scientific Notation

Definition of a Natural Number Exponent If b is a real number and n is a natural number, bn is read “the nth power of b” or “ b to the nth power.” Thus, the nth power of b is defined as the product of n factors of b. Furthermore, b1 = b “Special” powers are 2 (squared) and 3 (cubed). Find the exponent button on your calculator.

The Negative Exponent Rule If b is any real number other than 0 and n is a natural number, then The negative exponent “flips” the base to the other side of the division bar to become a positive exponent. It DOES NOT CHANGE the SIGN of the base! Ex: -2 3x-5 ans:

The Zero Exponent Rule If b is any real number other than 0, b0 = 1. Ex: (for x not equal to zero)

The Product Rule b m · b n = b m+n When multiplying exponential expressions with the same base, add the exponents. Use this sum as the exponent of the common base. Hint: if you get these rules confused, think of a simple example and work it manually. Such as:

The Power Rule (Powers to Powers) (bm)n = bm•n When an exponential expression is raised to a power, multiply the exponents. Place the product of the exponents on the base and remove the parentheses. Hint: if you get these rules confused, think of a simple example and work it manually. Such as:

The Quotient Rule When dividing exponential expressions with the same nonzero base, subtract the exponent in the denominator from the exponent in the numerator. Use this difference as the exponent of the common base. (Shortcut: subtract “up” or “down” depending on which is the smaller exponent.) Hint: if you get these rules confused, think of a simple example and work it manually. Such as: Q: Can you also think of an example that would demonstrate the zero exponent rule using the quotient rule?

Ex: Find the quotient a) b) 14x7 10x10 Ans: a) b)

Products to Powers (ab)n = anbn When a product is raised to a power, raise each factor to the power. Hint: if you get these rules confused, think of a simple example and work it manually.

Example Simplify: (-2y)4. =(-2y)(-2y)(-2y)(-2y) Solution Long way: (-2y)4 =(-2y)(-2y)(-2y)(-2y) =(-2)(-2)(-2)(-2)(y)(y)(y)(y) (by commutative law) = (-2)4y4 = 16y4 Short way: (-2y)4 = (-2)4y4 (by “products of powers”)

Now you try to simplify each of the following, then check below Now you try to simplify each of the following, then check below. (Hint: one of these cannot use any of the shortcut rules we have discussed, why?): -(-3x2y5)4 (x+2) 2 Ans: and The second one has ADDITION, our rules refer to mult or divisn.

Quotients to Powers When a quotient is raised to a power, raise the numerator to that power and divide by the denominator to that power.

Example Simplify by raising the quotient (15x7/6)-4 to the given power. Solution: Ans: (Hint: reduce inside parenthesis first!)

Properties of Exponents

Do problem #62 p 22. Ans: (Again, inside parenthesis first)

(optional) Scientific Notation The number 5.5 x 1012 is written in a form called scientific notation. A number in scientific notation is expressed as a number greater than or equal to 1 and less than 10 multiplied by some power of 10. It is customary to use the multiplication symbol, x, rather than a dot in scientific notation.

Example Write each number in decimal notation: a. 2.6 X 107 b. 1.016 X 10-8 Solution: a. 2.6 x 107 can be expressed in decimal notation by moving the decimal point in 2.6 seven places to the _______. 2.6 x 107 = ________________________. b. 1.016 x 10-8 can be expressed in decimal notation by moving the decimal point in 1.016 eight places to the _______. 1.016 x 10-8 = _______________________.

Scientific Notation To convert from decimal notation to scientific notation, we reverse the procedure. Move the decimal point in the given number to obtain a number greater than or equal to 1 and less than 10. The number of places the decimal point moves gives the exponent on 10; the exponent is positive if the given number is greater than 10 and negative if the given number is between 0 and 1. Example: Write each number in scientific notation. 4,600,000 ans: b. 0.00023 ans: