2LEARNING TARGETS AFTER YOU COMPLETE THIS CHAPTER, YOU WILL BE ABLE TO: DISTINGUISH RATIONAL AND IRRATIONAL NUMBERS.FIND ROOTS OF RADICALS.SIMPLIFY RADICAL EXPRESSIONS.ADD, SUBTRACT, MULTIPLY, AND DIVIDE RATIONAL EXPRESSIONS.RATIONALIZE A DENOMINATOR.SOLVE EQUATIONS AND USE A GRAPH TO FIND A SQUARE ROOT.
3CHAPTER VOCABULARYIRRATIONAL NUMBER: A NON-REPEATING AND NON-TERMINATING NUMBER.RATIONAL NUMBER: A NUMBER THAT CAN BE EXPRESSED AS A RATIO OF TWO NUMBERS: INCLUDES: 0 TO INFINITY ON WHOLE NUMBERS BOTH NEGATIVE AND POSITIVE, AND REGULAR FRACTIONS.DECIMAL EXPANSION: WRITING A NUMBER SUCH AS A FRACTION AS A DECIMAL.
4MORE CHAPTER VOCABULARY TERMINATING DECIMAL EXPANSION: A DECIMAL SEQUENCE THAT ENDS IN ZERO.REPEATING DECIMAL EXPANSION: A DECIMAL SEQUENCE WHERE THE NUMBERS REPEAT IN EXACTLY THE SAME ORDER.NON-TERMINATING: DOES NOT ENDNON-REPEATING: IT REPEATS BUT NOT IN A SET ORDER.
5EXAMPLES:IRRATIONAL NUMBER: pi, 22 divided by 7, the square root of 2 or 3.RATIONAL: 2, -3, ½ , ¼, the square root of 144.Decimal expansion: 2/5 = .40 – rationalDecimal expansion: 22/7 = … - irrational.Terminating: ¼ = .250Non-terminating: 7/11 =
6More Examples: Repeating: 1/9 = .111111111111111111 Non-repeating: 2/We mark repeating decimals with a line above the part that repeats:
7Rational Number Equivalents Rational Number Equivalents can be calculated using an equation. Let x = the expansion written as a repeating decimal. Multiply both sides that will allow the repeating digit to go on the left of the decimal (usually 10, 100, 1000). Subtract the repeating decimal and solve the one step equation for the rational number that is equal to a repeating decimal.
8Rational Number Equivalent Example:Take: ….Set up: 10 (x) = (10) etc10x = (use four places)-1x =9x = 3.000x = 1/3 so = 1/3
9Irrational Numbers as Decimals Radical: a number that is written with the radical sign.Radical sign:Sing for approximately equal to:
10Square roots and moreYou can find a square root of a number but also cube roots, 4th roots, 5th roots, etc.Square root of 36 is 6, the square root of 49 is 7.Some square roots are irrational numbers such as the square root of 2 which is …The square root of 11 is …which is an irrational number.
12Products and Quotients of Radicals You can multiply and find the quotients of radials.Let’s say you want to multiply the square root of 4 and the square root of 25. So you are really multiplying 2 times 5 = 10.If you are dividing for example: divide the square root of 64 by the square root of 144, you are really dividing 8 by 12 or 4/5.
15Radicals and Fractions Rationalize the Denominator: changing a fraction with an irrational number to an equivalent fraction with a rational number.Conjugate: A factor that when multiplied rationalizes (or simplifies) an expression.
16Radicals And Exponents Radicand: a number under the radical sign.Parts of a radical:
17Square Root Graph A way to find a square root using a graph. You will set up your graph with the squares of every number from 1 to 10 on the x-axis.Then on the y-axis choose a system for numbers counting by 5’s. (Go up to at least 100)When the graph and the number of the roots you are look for align, that is the square root of that number.