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Scientific Notation

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**What is scientific Notation?**

a way of expressing really big numbers or really small numbers in a more concise form.

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**N x 10x Consists of: Coefficient Base Exponent**

A number greater than 1 and less than 10 Base 10 Exponent “power” of 10 N x 10x

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Scientific Notation 6.02 x 106 m Standard form 6,020,000 m

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**To change standard form to scientific notation…**

Place the decimal point so that there is one non-zero digit to the left of the decimal point. Count the number of decimal places the decimal point has “moved” from the original number. This will be the exponent on the 10.

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Continued… If the original number in standard form was less than 1, then the exponent is negative. If the original number in standard form was greater than 1, then the exponent is positive.

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**Example 1 Convert 289,800,000 to scientific notation**

Place the decimal so there is one non-zero digit to its left : 289,800, Count the number of places the decimal was moved to determine exponent moved 8 places, so exponent is 8 Determine if exponent is + or – based on value of number in standard form Original number in standard form is greater than one, so exponent is + Final Answer: x 108

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**Example 2 Convert 0.000567 to scientific notation**

Place the decimal so there is one non-zero digit to its left : Count the number of places the decimal was moved to determine exponent moved 4 places, so exponent is 4 Determine if exponent is + or – based on value of number in standard form Original number in standard form is less than one, so exponent is - Final Answer: 5.67 x 10-4

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Practice Use the link below to practice converting standard form to scientific notation. Converting to Scientific Notation

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**To change scientific notation to standard form…**

Just move the decimal the number of places given by the exponent Positive exponents will result in numbers greater than one when in standard form So just think “which way do I move the decimal to make the resulting number greater than one?” To the right! Negative exponents will result in numbers less than one when in standard form So just think “which way do I move the decimal to make the resulting number less than one?” To the left! (Use zeros as placeholders!)

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**Example 3 Convert 5.093 x 106 to standard form.**

The exponent is 6, so move the decimal 6 places. The exponent is positive, meaning the result in standard form is greater than one. Decimal must be moved to the right to get a result greater than one. (note that three 0’s needed to hold places) Answer: 5,093,000 (moved 6 places to the right)

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**Example 4 Convert 1.976 x 10-4 to standard form.**

The exponent is 4, so move the decimal 4 places. The exponent is negative, meaning the result in standard form is less than one. Decimal must be moved to the left to get a result less than one. (note the 0’s added as placeholders) Answer: (moved 4 places to the left)

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Practice Use the link below to practice converting scientific notation to standard form Converting to Standard Form

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Even More Practice Below is a list of links to games and activities all having to do with scientific notation.

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**Now take the quiz to test your scientific notation skills!**

Click on the link below to take the quiz and then use the answer key for the correct answers. Quiz Answers

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**Using scientific notation in calculations: multiplication and division**

When multiplying numbers in scientific notation, multiply the coefficients and add the exponents. When dividing numbers in scientific notation, divide the coefficients and subtract the exponents.

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**Example 5 (2.0 x 105) x (1.5 x 10-2)= ? Multiply the coefficients**

Add the exponents 5 + (-2) = 3 (new exponent) Answer: 3.0 x 103

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**Example 6 (6.0 x 105) / (1.5 x 10-2)= ? Divide the coefficients**

6.0 / 1.5= 4.0 Subtract the exponents 5 - (-2) = 7 (new exponent) Answer: 4.0 x 107

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**Using scientific notation in calculations: addition and subtraction**

When adding or subtracting numbers in scientific notation, the exponents MUST MATCH (unless you punch it into a calculator!) The exponent on the final answer is the same as the exponent on the terms in the problem.

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**Changing the exponent (1.5 x 10-1) + (2.5 x 102) = ??**

The first thing we have to do is make our exponents match- they both either need to be -1 or 2, it doesn’t matter which, as long as they match, without changing the magnitude (value) of the quantity represented.

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1.5 x 10-1 = _____x 102 We are moving three powers of 10 to get from -1 to 2 (10-1 100 101 102) Think : 2 – (-1)= 3 So we need to move our decimal 3 places Now which direction? Since our original number is less than one, it has to stay less than one, so which way do we go? To the left!

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1.5 x 10-1 = x 102 Note that we moved in the positive direction to change the exponent from -1 to 2, and the negative direction to change the decimal.

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**Now we can add our numbers together**

(1.5 x 10-1) + (2.5 x 102) = ( x 102) + (2.5 x 102) = ( ) x 102= x 102

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**Let’s try to make both exponents -1…**

(1.5 x 10-1) + (2.5 x 102) = 2.5 x 102 =____ x 10-1 To get from 102 to 10-1, we must move three places in the negative direction. So we will move our decimal 3 places in the positive direction 2.5 x 102 =2500 x 10-1

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**Now we can add our numbers**

(1.5 x 10-1) + (2500 x 10-1) = ( ) x 10-1= x 10-1

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Note that our answer is not in correct form (one nonzero digit to left of decimal) so we need to put it in correct form: use the same logic Move decimal three places in negative direction, then move exponent three places in positive direction x 10-1 = x 102

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**So, to change an exponent…**

If you are going in the positive direction to get to the new exponent, move the decimal in the negative direction. If you are going in the negative direction to get to the new exponent, move the decimal in the positive direction.

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**A note about scientific notation and your calculator…**

On the graphing calculator, scientific notation is best done with the button. 4.58 x 106 is typed The other option is to use the 4.58 x But you have to be careful with this when dividing and make sure you use parantheses!

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**Use a calculator to evaluate: 4.5 x 10-5 1.6 x 10-2**

Type You must include parentheses if you don’t use the !! key but instead! (4.5 x ) (1.6 x ) = Write in scientific notation. = x 10-3

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DON’T FORGET!!! When you are copying an answer from your calculator screen make sure you include the scientific notation if present. If your calculator answer was 1.93E-2 you would copy down 1.93 x 10-2 on your paper. The “E” in the answer means “x 10”

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