Please turn in your Home-learning, get your notebook

and Springboard book, and begin the bell-ringer!

Test on Activity 6, 7 and 8 Wednesday (A day) and Thursday (B day).

Bell-Ringer #17 10/30/14

Write the following in either standard form or scientific notation.

1. 0.000006824

2. 3.71 x 100

Simplify the following. Write your answer in both scientific notation and standard form.

3. 7.1 x 105 – 3.26 x 105

Properties of Exponents and Scientific Notation10/30/14

Different forms:Exponential Form: 45 x 43 or

Expanded Form:

45 x 43 = (4x4x4x4x4)(4x4x4)

=

Standard Form:

45 x 43 = 65,536

= 625

Multiplying with ExponentsRULE: When multiplying two exponential expressions with the same base, you ADD the exponents.

Example: 59 x 53 = 59+3 =512

The bases are the same (5), therefore you add the two exponents (9+3).

Dividing with ExponentsRULE: When dividing two exponential expressions with the same base, you SUBTRACT the exponents.

Example: = 38-6 = 32

The bases are the same (3), therefore, you can subtract the exponents (8-6).

Negative Exponents When DIVIDING two exponential expressions that will result in an expression with a negative exponent, you have two options:

1. Divide by subtracting the exponents.

2. Write the numerator and denominator in expanded form and simplify.

=

HINT: You must know your rules for operations with integers in order to be able to successfully solve problems with negative exponents!

Exponent of 0 and 1Anything raised to the power of zero (0) is always one (1).

70 = 1

•Anything raised to the power of one (1) is always itself.

71 = 7

Powers of PowersWhen an exponential expression is raised to a power, you multiply the two exponents.

(88)6 = 88x6 =848

(1011)2 =1011x2 =1022

Rules for Operations with Integers Addition (when the signs are the same)

1. Keep the Sign2. Add

Addition (when the signs are different)1. Keep the sign of the number with the greater absolute

value.2. Subtract the bigger number from the smaller number.

Subtraction (Think L-C-O…Leave, Change, Opposite)1. Change the subtraction sign to addition sign.2. Change the sign of the second number.3. Follow Addition Rules.

Multiplication/Division1. Positive & Positive = Positive2. Negative & Negative = Positive3. Positive & Negative = Negative4. Negative & Positive = Negative

Examples

(-9) + (-4) =

(-7) + 4 =

8 – 11 =

15 – (-7) =

(-5) x (-9) =

(-90) ÷ 3=

Scientific NotationScientific Notation: a way to write a number as a product of the number, a, and 10n, when 1≤ a <10 (a needs to be at least equal to 1 but less than 10) and n is an integer.

a x 10n 5.21 x 106

Standard Form: a way to write a number using a digit for each place.

591,157.21

Convert from Scientific Notation to Standard Form: 5.12 x 106

Step 1: Simplify 106

106 = 10 x 10 x 10 x 10 x 10 x 10 = 1,000,000

Step 2: Multiply by 5.12

5.12 x 1,000,000 = 5,120,000

Hint: The exponent tells you how many spaces to move the decimal. When converting from scientific notation to standard form and the exponent is positive, you move the decimal to the RIGHT and fill spaces with zeroes.

Convert from Standard Form to Scientific Notation: 860,000

Step 1: Identify the location of the decimal point in 860,000. In all whole numbers, the decimal point is at the end (all the way to the right) of a number.

860,000. Decimal Point

Step 2: Move the decimal point to the left until you have a number that is greater than or equal to 1 and less than 10. Count the number of places you moved the decimal point.

860,000. the decimal is moved 5 places

Step 3: Rewrite the number in scientific notation. The number of places you moved the decimal is the exponent for the base of 10.

8.6 x 105

Remember: Scientific Notation requires that the value for “a” be at least 1 and less than 10.

Scientific Notation: Power of Zero, Negative Exponents, and Ordering

Power of Zero

4.5 x 100 = 4.5 x 1 = 4.5

Negative Exponents

Standard form to Scientific notation: 0.00000651 = 6.51 x 10-6

Count the number of paces the decimal is moved to the right to make the number between 1 and 10, the number of places moved to the right is written as the exponent for 10 and should be negative.

Scientific Notation to Standard form: 8.75 x 10-7 = 0.000000875

Move the decimal to the left according to the number in the exponent.

Ordering Use the values of the exponents to help determine the order. The smaller the exponent, the

smaller the value. The greater the exponent, the greater the value. Write the number in standard form to check the order. Write the numbers in order in their original form (scientific notation).

Scientific Notation: Estimation

To estimate:1. Look for the greatest place value and round to that place value. 2. Follow the rules for converting from standard form to scientific notation.

Examples: 0.00005146 456,145,956

Step 1:

Step 2:

Solution:

Multiplying in Scientific Notation

Step 1: use the commutative and associative properties of multiplication to regroup and reorder the multiplication problem.

Step 2: Multiply

Solution:

(2.15 x 108) x (1.24 x 103)

Step 1:

(2.15 x 1.24) x (108 x 103)

Step 2: 2.15 108+3 =1011

x1.24

2.666 x 1011

Standard form: 266,600,000,000

Dividing in Scientific Notation

Step 1: Divide the factors of a.

a x 10n a

a x 10n a

Step 2: Then, apply the rules for dividing exponential expressions. (you subtract the exponents)

n-n

Solution: Rewrite the answer in scientific notation using the number in step 1 and 2.

16.4 x 109

4.1 x 105

Step 1:

16.4 = 4

4.1

Step 2:

109-5 =104

4 x 104

Standard form: 40,000

Adding/Subtracting in Scientific Notation

Step 1: To add or subtract in scientific notation, the exponents must be the same. If they are not the same, rewrite the terms so that the exponents are the same. To do so, determine the number by which to increase the smaller exponent by so it is equal to the larger exponent.

Increase the smaller exponent by this number and move the decimal point of the number with the smaller exponent to the left the same number of places.

4.2 x 103 + 2.9 x 103 same exponents

Step 2: Add or subtract the new factors (digits) for “a.” 4.2 + 2.9 = 7.1

Solution: Write the sum in scientific notation. If the answer is not in scientific notation (i.e. if “a” is not between 1 and10 ) convert it to scientific notation.

7.1 x 103

Examples:

8.5 x 108 – 6.2 x 106

2.1 x 104 + 3.6 x 105