- WELCOME TO A SECOND LESSON ON SIMPLIFYING RADICAL EXPRESSIONS.
IN THIS LESSON, WE'LL SIMPLIFY RADICAL EXPRESSIONS
WITH FRACTIONS.
RADICAL EXPRESSION IS SAID TO BE IN SIMPLEST FORM
IF ALL OF THE FOLLOWING THREE STATEMENTS ARE TRUE.
NUMBER 1, ALL POSSIBLE nth POWERED FACTORS
HAVE BEEN REMOVED FROM EACH RADICAL,
AND THIS WAS THE MAIN FOCUS OF OUR FIRST LESSON.
NUMBER 2, NO RADICAL CONTAINS A FRACTION.
NUMBER 3, NO DENOMINATOR OF A FRACTION CONTAINS A RADICAL.
IF THE DENOMINATOR DOES CONTAIN A RADICAL,
WE'LL MULTIPLY THE NUMERATOR AND THE DENOMINATOR BY A RADICAL
THAT MAKES THE DENOMINATOR A PERFECT nth ROOT.
MEANING IT WILL SIMPLIFY PERFECTLY
AND THIS PROCESS IS CALLED RATIONALIZING THE DENOMINATOR.
LET'S TAKE A LOOK AT OUR FIRST EXAMPLE.
HERE WE HAVE THE SQUARE ROOT OF 4/9.
THIS IS NOT IN SIMPLEST FORM
BECAUSE WE HAVE A FRACTION UNDER THE SQUARE ROOT.
SO FOR THE FIRST STEP,
WE'RE GOING TO TAKE ADVANTAGE OF THIS PROPERTY HERE THAT SAYS
IF WE HAVE THE SQUARE ROOT OF A FRACTION
IT'S EQUAL TO THE SQUARE ROOT OF THE NUMERATOR
DIVIDED BY THE SQUARE ROOT OF THE DENOMINATOR,
WHICH MEANS THE SQUARE ROOT OF 4/9
IS EQUAL TO THE SQUARED OF 4 DIVIDED BY THE SQUARED OF 9.
NOW THIS IS NOT IN SIMPLEST FORM
BECAUSE NOW THE DENOMINATOR CONTAINS A RADICAL.
BUT NOTICE HOW BOTH 4 AND 9 ARE PERFECT SQUARES
SO THIS SIMPLIFIES PERFECTLY
AND THE REASON IT DOES IS BECAUSE 4 IS A PERFECT SQUARE
SINCE WE CAN WRITE 4 AS 2 x 2
AND 9 IS ALSO A PERFECT SQUARE BECAUSE WE CAN WRITE 9 AS 3 x 3.
SO BECAUSE THESE ARE SQUARE ROOTS MEANING THE INDEX IS 2
AND BOTH SQUARE ROOTS HAVE TWO EQUAL FACTORS
THIS SIMPLIFIES PERFECTLY.
THE SQUARED OF 4 OR THE SQUARED OF 2 x 2 = 2
AND THE SQUARED OF 9 OR THE SQUARED OF 3 x 3 = 3.
SO NOW THE RADICAL IS SIMPLIFIED.
BUT THEY WON'T ALL BE THIS STRAIGHT FORWARD.
LET'S TAKE A LOOK AT A SECOND EXAMPLE.
HERE WE HAVE THE SQUARED OF 9/11
AND AGAIN THE FIRST STEP IS TO BREAK THIS UP
INTO THE SQUARED OF THE NUMERATOR
AND THE SQUARED OF THE DENOMINATOR.
SO WE HAVE THE SQUARE ROOT OF 9/11,
WHICH IS EQUAL TO THE SQUARED OF 9 DIVIDED BY THE SQUARED OF 11.
NOW WE KNOW THE NUMERATOR DOES SIMPLIFY PERFECTLY.
WE WANT TO FOCUS ON THE DENOMINATOR
BECAUSE THIS IS NOT IN SIMPLEST FORM
BECAUSE WE HAVE A SQUARE ROOT IN THE DENOMINATOR.
SO IF WE WANT THE DENOMINATOR TO SIMPLIFY PERFECTLY,
NOTICE HOW WE'D HAVE TO HAVE 2 FACTORS OF 11
UNDERNEATH THE SQUARE ROOT IN THE DENOMINATOR.
AND THIS IS WHY WE'RE GOING TO MULTIPLY
BOTH THE NUMERATOR AND THE DENOMINATOR BY SQUARED OF 11.
SO KNOW WHEN WE FIND THIS PRODUCT,
THE NUMERATOR'S GOING TO BE THE SQUARE ROOT OF--
WELL 9 = 3 x 3 AND 11 IS PRIME.
SO WE HAVE THE SQUARED OF (3 x 3) x 11
AND NOW IN THE DENOMINATOR,
WE WOULD HAVE THE SQUARE ROOT OF 11 x 11.
NOTICE HOW NOW WE HAVE A PERFECT SQUARE
UNDERNEATH THE SQUARE ROOT AND DENOMINATOR,
WHICH WILL SIMPLIFY PERFECTLY.
SO NOW WE'LL SIMPLIFY THIS.
LOOKING AT THE DENOMINATOR, THIS WILL SIMPLIFY.
LOOKING AT THE NUMERATOR THIS WILL SIMPLIFY.
SO IN SIMPLIFIED FORM THIS WILL BE EQUAL 3 SQUARED OF 11
DIVIDED BY 11.
THIS IS CONSIDERED THE SIMPLIFIED FORM
OF SQUARE ROOT 9/11.
IT DOESN'T REALLY LOOK ANYMORE SIMPLIFIED
BUT THE MAIN THING WE SHOULD NOTICE IS THAT
THE DENOMINATOR DOES NOT CONTAIN A RADICAL
AND THERE'S NO FRACTION UNDERNEATH THE RADICAL.
NOW LET'S TAKE A LOOK AT THE CUBE ROOT OF 2/3.
AGAIN, THIS IS NOT CONSIDERED SIMPLIFIED
BECAUSE WE HAVE A FRACTION UNDER THE RADICAL.
SO THE FIRST STEP, WE'RE GOING TO WRITE THIS
AS THE CUBE ROOT OF 2 DIVIDED BY THE CUBE ROOT OF 3.
NOW THIS IS NOT CONSIDERED SIMPLIFIED
BECAUSE WE HAVE A RADICAL IN THE DENOMINATOR.
NOW WE WANT TO FOCUS ON WHAT IT'S GOING TO TAKE
TO MAKE THE DENOMINATOR SIMPLIFY PERFECTLY.
WELL NOW BECAUSE WE HAVE A CUBE ROOT,
IF WE HAD THREE FACTORS OF 3
THIS WOULD SIMPLIFY PERFECTLY
WHICH MEANS WE'RE GOING TO MULTIPLY
BOTH THE NUMERATOR AND DENOMINATOR
BY THE CUBE ROOT OF 3 x 3.
NOTICE IN OUR DENOMINATOR WE WOULD NOW HAVE
THREE FACTORS OF 3 AND SINCE WE HAVE A CUBE ROOT
IT'S GOING TO SIMPLIFY PERFECTLY.
BUT OF COURSE WE HAVE TO DO THE SAME TO THE NUMERATOR.
SO WE'LL HAVE THE CUBE ROOT OF 3 x 3.
NOW LET'S GO AHEAD AND MULTIPLY.
LOOKING AT THE NUMERATOR,
WE WOULD NOW HAVE THE CUBE ROOT OF (2 x 3) x 3
AND THE DENOMINATOR WOULD NOW BE THE CUBE ROOT OF (3 x 3) x 3,
AND AGAIN, BECAUSE WE HAVE A CUBE ROOT
WE'RE LOOKING FOR GROUPS OF THREE EQUAL FACTORS
WHICH WOULD BE PERFECT CUBES WHICH WE HAVE HERE.
NOTICE IN THE NUMERATOR WE DON'T HAVE THREE EQUAL FACTORS
SO THE NUMERATOR WOULD NOT SIMPLIFY.
SO THE SIMPLIFIED FORM,
WE WOULD HAVE THE CUBE ROOT OF (2 x 3) x 3
THAT'S GOING TO BE 18.
AND THE DENOMINATOR SIMPLIFIES PERFECTLY TO 3.
AGAIN, IT DOESN'T LOOK ANYMORE SIMPLIFIED
BUT BY DEFINITION THIS IS THE SIMPLIFIED FORM
OF CUBE ROOT 2/3.
NOW LET'S TAKE A LOOK AT TWO MORE EXAMPLES
THAT CONTAIN VARIABLES.
HERE WE HAVE THE SQUARED OF 2 DIVIDED BY 5X SQUARED Y THE 3rd.
FIRST STEP IS TO WRITE THIS AS THE SQUARE ROOT OF 2
DIVIDED BY THE SQUARE ROOT OF 5--
I'M GOING TO WRITE X SQUARED
AS X x X AND I'M GOING TO WRITE Y TO THE 3rd AS (Y x Y) x Y
AND AGAIN, OUR GOAL HERE IS TO MAKE THE DENOMINATOR
SIMPLIFY PERFECTLY MEANING WE WANT GROUPS OF TWO EQUAL FACTORS
FOR EVERY EXISTING FACTOR IN THE DENOMINATOR.
SO TO MAKE THE DENOMINATOR SIMPLIFY PERFECTLY,
NOTICE HOW WE ONLY HAVE ONE FACTOR OF 5.
WE'D HAVE TO HAVE TWO TO MAKE IT SIMPLIFY PERFECTLY.
SO WE'RE GOING TO MULTIPLY BY THE SQUARED OF ONE FACTOR OF 5.
NOTICE HOW WE ALREADY HAVE TWO FACTORS OF X.
THAT WILL SIMPLIFY PERFECTLY.
WE ALREADY HAVE TWO FACTORS OF Y
BUT TO MAKE THIS SIMPLIFY PERFECTLY
NOTICE HOW WE NEED ONE MORE FACTOR OF Y
SO WE'RE GOING TO MULTIPLY IT BY THE SQUARED OF 5Y.
AND THEN WE DO THE SAME TO THE NUMERATOR.
SO WE HAVE THE SQUARED OF 5Y HERE AND NOW LET'S MULTIPLY.
IN THE NUMERATOR WE HAVE THE SQUARE ROOT OF (2 x 5) x Y.
OUR DENOMINATOR WILL NOW BE THE SQUARE ROOT OF 5 x 5.
WE STILL HAVE TWO FACTORS OF X
AND NOW WE HAVE 1, 2, 3, 4, FACTORS OF Y.
AND AGAIN, BECAUSE WE HAVE A SQUARE ROOT
MEANING THE INDEX IS 2 WE'LL MAKE GROUPS OF 2
FOR THE EQUAL FACTORS.
SO HERE WE HAVE TWO FACTORS OF 5, TWO FACTORS OF X,
TWO FACTORS OF Y, AND TWO FACTORS OF Y.
AND AGAIN, THE KEY HERE IS NOTICING THE DENOMINATOR
IS GOING TO SIMPLIFY PERFECTLY.
NOTICE THE NUMERATOR IS NOT GOING TO SIMPLIFY.
SO WE'LL HAVE THE SQUARE ROOT OF 10Y DIVIDED BY--
OUR DENOMINATOR IS NOW GOING TO BE 5,
ONE FACTOR OF X AND TWO FACTORS OF Y, ONE FROM EACH GROUP.
SO WE'D HAVE Y SQUARED.
THERE'S ONE MORE THING TO MENTION ABOUT THIS EXAMPLE.
REMEMBER IN THE PREVIOUS VIDEO WE SAID,
WHEN SIMPLIFYING RADICALS WITH VARIABLES,
IF THE INDEX IS EVEN
AND THE SIMPLIFIED VARIABLE HAS AN ODD EXPONENT,
LIKE HERE WE HAVE X TO THE 1st,
WE DO NEED TO INCLUDE THAT VARIABLE PART
IN AN ABSOLUTE VALUE TO ASSURE THE RESULT IS POSITIVE.
SO LET'S GO AHEAD AND WRITE THIS AS THE SQUARE ROOT OF 10Y.
WE CAN PUT JUST THE X IN THE ABSOLUTE VALUE
OR THE ENTIRE DENOMINATOR.
LET'S GO AND JUST PUT THE X THIS TIME.
SO WE HAVE 5 ABSOLUTE VALUE OF XY SQUARED.
NOW IF WE WERE TOLD THE VARIABLES WERE ALWAYS POSITIVE
WE WOULD NOT HAVE TO INCLUDE THE ABSOLUTE VALUE,
AND AGAIN, IT ONLY COMES INTO PLAY WHEN THE INDEX IS EVEN
AND THE SIMPLIFIED VARIABLE HAS AN ODD EXPONENT.
LET'S LOOK AT ONE MORE.
HERE WE HAVE THE CUBE ROOT OF 5 DIVIDED BY 4X SQUARED Y.
FIRST STEP, WRITE IT AS THE CUBE ROOT OF 5,
DIVIDED BY THE CUBE ROOT--I'M GOING TO WRITE 4 AS 2 x 2,
TWO FACTORS OF X, AND ONE FACTOR OF Y.
AGAIN, FOCUSING ON THE DENOMINATOR
WHICH WE WANT TO SIMPLIFY PERFECTLY
WE WOULD HAVE TO HAVE GROUPS OF THREE EQUAL FACTORS NOW
BECAUSE THE INDEX IS 3.
SO LOOKING AT THE DENOMINATOR,
NOTICE HOW WE HAVE TWO FACTORS OF 2,
BUT WE NEED THREE TO MAKE IT SIMPLIFY PERFECTLY.
SO WE'LL GOING TO MULTIPLY IT BY THE CUBE ROOT OF 2.
WE HAVE TWO FACTORS OF X, WE NEED ONE MORE X.
AND WE ONLY HAVE ONE FACTOR OF Y.
WE NEED TWO MORE FACTORS OF Y.
NOTICE NOW WE HAVE THREE OF EACH FACTOR.
WE'LL DO THE SAME TO THE NUMERATOR.
NOW WE'LL MULTIPLY.
IN THE NUMERATOR WE HAVE (2 x 5) x (X x Y) x Y.
NOTICE NOTHING'S GOING TO SIMPLIFY.
BUT IN THE DENOMINATOR, WE NOW HAVE THREE FACTORS OF 2,
THREE FACTORS OF X AND THREE FACTORS OF Y, WHICH IS PERFECT
BECAUSE THE INDEX IS 3.
HERE'S A PERFECT CUBE FACTOR AS WELL AS HERE AND HERE.
NOW IN SIMPLIFIED FORM WE'D HAVE THE CUBE ROOT OF 10XY SQUARED
DIVIDED BY--WE'D HAVE 2XY.
AND WE NEVER HAVE TO WORRY ABOUT THE ABSOLUTE VALUE
IF THE INDEX IS ODD.
I HOPE YOU FOUND THESE EXAMPLES HELPFUL.