1. { Scientific Notation Measuring the very largest distances and the very smallest portions…

2.  In order to understand “scientific notation” we first need to understand exponents. Most of us understand the concept – because we know the numbers 1 – 12 “squared.” 1 2 = 12 2 = 43 2 = 9 4 2 = 165 2 = 256 2 = 36 7 2 = 498 2 = 649 2 = 81 10 2 = 10011 2 = 12112 2 = 144 Understanding Exponents

3.  We can figure these in reverse as well! This is taking the square root of a number! √144 = ____√121 = ____√81 = ____ √100 = ____√1 = ____√4 = ____ √9 = ____√25 = ____√36 = ____ √16 = ____√64 = ____√49 = ____ Square Roots

4. { You should know by now that a number “squared” is just a number being multiplied by itself! Consider the equation: 2 2 = 4 2 2 = 4 We know that this is just another way to state: 2 x 2 = 4. So what about this slightly different equation:? base exponent 2 3 Other Exponential Forms

5. So what is the value of 2 3 ? Exponent Base 2 3

6. {{ Two to the third power = Two to the third power = 23 =23 = 2 x 2 x 2 = 8 Two cubed equals eight.

7. What other examples can we solve involving cubed numbers? A. 1 3 = 1 x 1 x 1 = _____E. 6 3 = 6 x 6 x 6 = _____ A. 1 3 = 1 x 1 x 1 = _____E. 6 3 = 6 x 6 x 6 = _____ B. 3 3 = 3 x 3 x 3 = _____F. 7 3 = 7 x 7 x 7 = _____ B. 3 3 = 3 x 3 x 3 = _____F. 7 3 = 7 x 7 x 7 = _____ C. 4 3 = 4 x 4 x 4 = _____G. 8 3 = 8 x 8 x 8 = _____ C. 4 3 = 4 x 4 x 4 = _____G. 8 3 = 8 x 8 x 8 = _____ D. 5 3 = 5 x 5 x 5 = _____H. 10 3 = 10 x 10 x 10 = D. 5 3 = 5 x 5 x 5 = _____H. 10 3 = 10 x 10 x 10 = …and we can keep this up all day!

8. Any number to the power of zero is equal to one. This is true no matter how larger or small a number is – and no matter whether the number is positive or negative. Example 1. 587 0 = 1 Example 2. 55 0 = 1 Example 3.1 0 = 1 The only exception to the rule would be 0 0, because zero to the zero power is undefined. It doesn’t exist. Special Exponents: n 0

9. Any number to the power of one is equal to the number! This is true no matter how larger or small a number is – and no matter whether the number is positive or negative. Example 1. 587 1 = 587 Example 2. 55 1 = 55 Example 3.1 1 = 1 Special Exponents: n 1

10. Solve these equations with exponents! A. 2343 0 = _____B. 345 1 = _____ C. -258 0 = _____D. -854 1 = _____ E. 265 1 = _____F. 0 1 = _____ G. 5 3 = _____H. 2 4 = _____ Practice with exponents!

11. The number ten is a very important one in mathematics – and for many reasons. Our system of counting is a base ten system. Meaning that the concept of “place value” in our counting is achieved by advancing in units of ten! For example, ten units of one = 10. And ten units of ten = 100. Ten units of 100 = 1000; ten units of 1000 = 10, 000. And so on, infinitely! This is place value. And you understand it, right? 10 to Exponential Powers, or 10 n

12. Here are some very simple examples: A. 7, or =10 B. 45, or =100 C. 659, or =1,000 D. 10,000, or =1, 000 E. 89,899, or =100,000 Examples of place value!

13.  You know that 7 is less than 10, even though the number seven is larger than both 1 an 0 – or even 1 and 0 combined.  You know that 10,000 is greater than 1,000 even though the numbers involved are essentially the same.  And you know that 89, 899 is still less than 100,000 – even though every number in 89, 899 is larger than the 1 and zeroes in 100,000! We know about place value!

14.  Scientific notation is simply another way to measure place value. We use scientific notation in two basic contexts!  1. When we are using extremely large numbers!  OR  2. When we are using infinitesimally small numbers! Scientific Notation

15. { Consider this example: What is the distance from the Earth to the Sun in miles? The answer is approximately 93 Million miles! We can write this out longhand – 93, 000, 000 miles. Or, we can abbreviate the number using scientific notion. The Distance from the Earth to the Sun!

16. { The Earth is 93,000,000 miles from the Sun. 9.3 X 10 7 miles from the Sun.

17. Because our system of place value is base ten, we can easily measure large numbers – and smaller numbers, too – by using our knowledge of the number ten’s exponential values! CHECK IT! 10 0 = 110 5 = 100000 10 1 = 1010 6 = 1000000 10 2 = 10010 7 = 10000000 10 3 = 100010 8 = 100000000 10 4 = 1000010 9 = 1000000000 Ten to the n th power! And we can do this for any power of 10… Infinitely!

18.  Consider these examples: A. The distance between the Sun and the planet Jupiter : 483, 700, 000 miles. B. The number of people on the planet Earth. Total population: 6, 960, 000, 000. Representing large numbers in Scientific Notation.

19.  Since we all know the value of 10 n, we are able to use exponents of ten to represent the place value of large numbers.  The distance between the sun and the planet Jupiter, then, becomes this multiplication product: 4.837 x 10 8.  We know the value of 10 8 is 100, 000, 000. And the “significant figures” or “sig figs” in the expression are used to create a “shorthand” multiplication problem.  4.837 x 100, 000, 000 = 483, 700, 000. Numbers in Scientific Notation.

20.  Since we all know the value of 10 n, we are able to use exponents of ten to represent the place value of large numbers.  The population of the planet Earth, approximately 6.96 billion people, or 6, 960, 000, 000 becomes this multiplication product: 6.96 x 10 9.  We know the value of 10 9 is 1, 000, 000, 000. And the “significant figures” or “sig figs” in the expression are used to create a “shorthand” multiplication problem.  6.96 x 1,000, 000, 000 = 6, 960, 000, 000 or 6.96 Billion! The World Population in Scientific Notation.

21.  A shorter method of writing numbers in scientific notation is to identify the exponent of 10 in the number and literally move the decimal by that number of “places.”  Consider these examples. Note that the.0 at the end of each number does not change it’s value at all! 1.0 = 1, right? A. 1.000 x 10 3 = 1, 000.0 B. 7.55 x 10 6 = 7, 550, 000.0 C. 3.65 x 10 21 = 3, 650, 000, 000, 000, 000, 000, 000.0 When we multiply by ten…

22.  Write each of the numbers below in Scientific Notation: A. 7,000, 000, 000 B. 8, 500, 000 C. 5,000D. 25, 000, 000, 000 E. 63, 000, 000F. 9, 600 G. The United States of America’s current national debt: $14, 700, 000, 000, 000. national debt: $14, 700, 000, 000, 000. (Yes, you need to use scientific notation for that!) Practice? We talkin’ about practice?