Naturals, Integers, Fractions, Decimals, Rationals, Square Roots, Reals, Formulas, Scientific Notation, Graphing
Irrational Numbers - Closure is an important principle in algebra. For instance,the natural numbers are closed under addition. This means that a natural number added to another natural number gives us a natural number. Closure allows us to deduce some simple properties of numbers. Natural numbers are not closed under subtraction (you may get zero or a negative integer). Integers are not closed under division. Previously, we learned that rational numbers (whole numbers and fractions, some of them improper fractions) are not closed under square roots. This complicates the situation, because it shows us that we have a whole new set of numbers, the irrational numbers. It turns out that the set of irrational numbers vastly outnumbers the set of rational numbers.
Irrational numbers are numbers that cannot be expressed as a fraction (proper or improper). Irrational numbers can be expressed in decimal form, but it is impossible to write them out exactly this way. In mathematics, a name can be used with a very precise meaning that may have little to do with the meaning of the English word. ("Irrational" numbers are NOT numbers that can't argue logically!)
Pi is an irrational number because it cannot be expressed as a ratio (fraction) of two integers: it has no exact decimal equivalent, although 3.1415926 is good enough for many applications. The square root of 2 is another irrational number that cannot be written as a fraction. In fact many irrational numbers arise from the square root operation.
Properties of Square Roots -
These properties allow us to rewrite many square roots in a simpler form.
Complete the following problems on a separate sheet of paper. Be sure to show all of the work necessary to complete the problems. (Calculators may be used to check the answer, but should not be used to solve the problem originally.)
Simplest Square Root Form
1. Change the square root of 27 to its simplest form.
Step 1 - Express 27 as a product of 9 times 3.
Step 2 - Use Property 4 to rewrite the square root of 27 as the product of the square root of 9 and the square root of 3.
Step 3 - Write the square root of 9 as 3 to get the final answer.
Example:
HOMEWORK PROBLEMS :
2. Change the square root of 8 to its simplest form.
3. Change the square root of 18 to its simplest form.
4. Change the square root of 48 to its simplest form.
5. Change the square root of 112 to its simplest form.
6. Find the decimal equivalent of the square root of 27 and its simplest form.
7. Find the decimal equivalent of the square root of 8 and its simplest form.
8. Find the decimal equivalent of the square root of 18 and its simplest form.
9. Find the decimal equivalent of the square root of 48 and its simplest form.
10. Find the decimal equivalent of the square root of 112 and its simplest form.
Adding and Subtracting Square Roots
11.
Simplify 3Ö 8 + 2Ö 18 - 4Ö 2Step 1 - Express each square root in simplest form.
Step 2 - Combine similar square root terms to get the final answer.
Example:
HOMEWORK PROBLEMS :
12. Simplify 3Ö 8 - Ö 32
13. Simplify 6Ö 12 + Ö 3 - 2Ö 48
14. Simplify 3Ö 5 - Ö 80
15. Simplify 13Ö 28 - 2Ö 63 - 7Ö 7
16. Verify that 3Ö 8 + 2Ö 18 - 4Ö 2 and 8Ö 2 are identical when expressed in decimal form.
17. Verify that 3Ö 8 - Ö 32 and its simplest form are identical when expressed in decimal form.
18. Verify that 6Ö 12 + Ö 3 - 2Ö 48 and its simplest form are identical when expressed in decimal form.
19. Verify that 3Ö 5 - Ö 80 and its simplest form are identical when expressed in decimal form.
20. Verify that 13Ö 28 - 2Ö 63 - 7Ö 7 and its simplest form are identical when expressed in decimal form.
Multiplying Square Roots
21. Multiply and s
implify (2Ö 8)x(3Ö 5)Step 1 - Express each square root in simplest form.
Step 2 - Multiply all the rational numbers together.
Step 3 - Multiply the numbers under the square root signs together.
Example:
HOMEWORK PROBLEMS :
22. Multiply and simplify (3Ö 8)x(Ö 32)
23. Multiply and simplify (6Ö 12)x(Ö 48)
24. Multiply and simplify (3Ö 5)x(Ö 80)
25. Multiply and simplify (13Ö 28)x(Ö 63)
26. Express (2Ö 8)x(3Ö 5) in decimal form.
27. Express (3Ö 8)x(Ö 32) in decimal form.
28. Express (6Ö 12)x(Ö 48) in decimal form.
29. Express (3Ö 5)x(Ö 80) in decimal form.
30. Express (13Ö 28)x(Ö 63) in decimal form.
Dividing Square Roots
31. Divide
6Ö 8 by Ö 5.Step 1 - Write the problem in fractional form and express all square roots in simplest form.
Step 2 - Multiply the numerator and the denominator by the square root in the denominator.
Step 3 - Cancel out any common factors between the numerator and the denominator.
Example:
HOMEWORK PROBLEMS :
32. Divide 3Ö 8 by Ö 32.
33. Divide 6Ö 12 by Ö 48.
34. Divide 3Ö 5 by Ö 80.
35. Divide 13Ö 28 by Ö 63.
36. Express (6Ö 8)/(Ö 5) in decimal form.
37. Express (3Ö 8)/(Ö 32) in decimal form.
38. Express (6Ö 12)/(Ö 48) in decimal form.
39. Express (3Ö 5)/(Ö 80) in decimal form.
40. Express (13Ö 28)/(Ö 63) in decimal form.