2 Learning Objective Evaluate Square Roots of real numbers Recognize that not all square roots represents real numbersDetermine whether the square root of a real number is rational or irrationalWrite square roots as exponential expressions
3 Key Vocabulary square root root principal square root imaginary radical signradicandradical expressionindexrootimaginaryperfect squareperfect square factorrational numberirrational number
4 Evaluate Square Roots of Real Numbers positive or principal square root uses the to indicate a positive square root if“The square root of a”Negative square roots are indicated by
5 Evaluate Square Roots of Real Numbers radical sign isradicand is the number under the radical signradical expression is the entire expressionIndex tells the root of the expression and the squared root index are not writtenAll squares of a nonzero real number must be positive
6 Evaluate Square Roots of Real Numbers square root is the reverse process of squaring a numberExample : because 72 = (7)(7) = 49The is 0, written
9 Negative Square Roots Negative square roots are not real numbers How do we know that the square of any nonzero real number must be a positive number? Example:
10 Perfect SquaresThe numbers 1, 4, 9, 16, 25, 36, 49, … are perfect squares because each number is a square of a natural number.See page 536 for a list of the first 20 perfect squares.Natural numbersSquare Natural numberPerfect squares
11 Rational NumbersA rational number can be written as a and b are integers and b ≠ 0Rational numbers written as a decimal are either terminating or repeating. ½ = or ⅓ = .333…Round you answers two decimal place and use the approximately equal sign ≈
12 Rational NumbersThe square root of every perfect square is also a rational number.
13 Irrational NumbersA irrational number is any number that is not rational and are non-terminating and non-repeating decimals.The square root of non perfect square are irrational number
14 Writing a Square Root in Exponential Form Reviewing the rules for exponents in chapter 4 section 1 and 2 may be helpful.
16 Review of Rules for Exponents Product Rule:Quotient Rule:Zero Exponent Rule:
17 Review of Rules for Exponents Power Rule:Expanded Power Rule:Negative Exponent Rule:
18 RememberThe square root is the opposite or reverse process of squaring.“Square of a number”“Multiply the number by itself”Every real number greater than 0 has two squares one positive and one negativeSquare roots of negative numbers are not real numbers. They are imaginary numbers
19 Remember The results of a square root is always nonnegative. The result is only rational if the radicand is a perfect squareThe radical form is the exact value.Calculators only give approximate values for irrational numbers. We use the ≈ sign for these values.
20 RememberYou should try to memorize as many perfect squares as possible to help with simplifying in the next section.Review of factoring may also be helpful for simplifying in the next section.Extra practice may be helpful to remember the difference between and