Presentation on theme: "The Radical Square Root"— Presentation transcript:
1The Radical Square Root The square root of any real number is a number, rational or irrational, that when multiplied by itself will result in a product that is the original numberThe RadicalRadical signSquare RootRadicandEvery positive radicand has a positive and negative sq. root.The principal Sq. Root of a number is the positive sq. root.A rational number can have a rational or irrational sq. rt.An irrational number can only have an irrational root.
2= 7.9 = 232 225 = +15 529 = +23 Model Problems Find to the nearest tenth:= 13.4= 7.9= 11.4= 232= 64.4Find the principal Square Root:225= +15529= +23Simplify:= |x|= x= 2x8= x + 1
3Index of 2 Square Root Index of 2 radical sign radicand index of a number is one of the twoequal factors whose product isthat numberSquare RootIndex of 2has an index of 2Every positive real numberhas two square rootsThe principal square root ofa positive number k is itspositive square root,If k < 0, is an imaginarynumber
4Index of 3 Cube Root Index = 3 radical sign radicand index of a number is one of the threeequal factors whose product isthat numberhas an index of 3principal cube roots
5nth RootThe nth root of a number (where n is any counting number) is one of n equal factors whose product is that number.k is the radicandn is the indexis the principal nth root of k25 = 32(-2)5 = -3254 = 625
6Index of n nth Root Index of n radical sign radicand index of a number is one of nequal factors whose product isthat numbernth RootIndex of nhas an index where n is anycounting numberprincipal odd rootsprincipal even roots
8In general, for non-negative numbers a, b and n Radical Rule #1In general, for non-negative numbers a, b and nExample:simplified= x4= x3Hint: will the index divide evenlyinto the exponent of radicand term?
9In general, for non-negative numbers a, b, and n Radical Rule #2True or False:IfandTTTransitive Propertyof EqualityIf a = b, and b = c,then a = cthenIn general, for non-negative numbers a, b, and nExample:
10Perfect Squares – Index 2 12144111211001098186474963652541639421
11Perfect Square Factors Find as many combinations of 2 factors whose product is 75Factors that are Perfect SquaresFind as many combinations of 2 factors whose product is 128
12Find as many combinations of 2 factors whose product is 80 Simplifying RadicalsSimplify:answer must be in radical form.Find as many combinations of 2 factors whose product is 80perfectsquarecomes outfrom underthe radicalTo simplify a radical find, if possible, 2 factors of the radicand, one of which is the largest perfect square of the radicand.The square root of the perfect square becomes a factor of the coefficient of the radical.
14Simplifying Radicals Simplify: answer must be in radical form. 1) Factor the radicand so that the perfect power (cube) is a factor2) Express the radical as the product of the roots of the factors3) Simplify the radical containing the largest perfect power (cube)
15Simplifying Radicals Simplify: 1) Change the radicand to an equivalent fraction whose denominator is a perfect power.2) Express the radical as the quotient of two roots3) Simplify the radical in the denominator
16Simplify: Model Problems KEY: Find 2 factors - one of which is the largest perfect square possible