- WE WANT TO SIMPLIFY THE GIVEN RADICAL EXPRESSIONS
INVOLVING SQUARE ROOTS
AND WE'LL SEE EXPRESSIONS LIKE THIS
WHEN USING THE QUADRATIC FORMULA.
THE FIRST STEP IS GOING TO BE TO SIMPLIFY THE SQUARE ROOT.
SO IN THIS CASE WE WANT TO SIMPLIFY THE SQUARE ROOT OF 32.
REMEMBER THIS WILL SIMPLIFY
IF IT CONTAINS ANY PERFECT SQUARE FACTORS.
WELL 32 IS = 16 x 2 AND 16 A PERFECT SQUARE.
THE SQUARE ROOT OF 16 = 4
SO THIS SIMPLIFIES TO 4 SQUARE ROOT 2.
REMEMBER IF WE DON'T RECOGNIZE OUR PERFECT SCORE FACTORS
WE CAN ALWAYS WRITE OUT THE PRIME FACTORIZATION OF 32.
THE PRIME FACTORIZATION OF 32 IS ACTUALLY 5 FACTORS OF 2.
EVERY TIME WE HAVE TWO EQUAL FACTORS
WE HAVE A PERFECT SQUARE.
SO HERE'S A PERFECT SQUARE AND HERE'S A PERFECT SQUARE.
REMEMBER TO SIMPLIFY THIS WE TAKE ONE FACTOR FROM EACH GROUP.
SO THIS SIMPLIFIES TO TWO FACTORS OF 2 OUTSIDE
WHICH EQUALS 4
AND WE'RE LEFT WITH THE SQUARE ROOT OF 2.
EITHER WAY IT SIMPLIFIES TO 4 SQUARE ROOT 2.
SO THEY CAN WRITE THIS AS 8 - 4 SQUARE ROOT 2
DIVIDED BY 4.
NOW WE NEED TO BE CAREFUL ON THIS NEXT STEP.
WE CANNOT JUST SIMPLIFY THIS 8 AND THIS 4
BECAUSE WE CANNOT SIMPLIFY ACROSS
IN THIS CASE, SUBTRACTION.
SO THERE ARE TWO WAYS TO SIMPLIFY THIS CORRECTLY.
ONE WAY SINCE WE'RE DIVIDING BY A BINOMIAL
WE SHOULD DIVIDE EACH TERM IN THE NUMERATOR BY 4.
SO WE CAN WRITE THIS AS 8 DIVIDED BY 4 - 4 SQUARE ROOT 2
DIVIDED BY 4.
WELL 8 DIVIDED BY 4 IS = 2
AND HERE 4/4 SIMPLIFIES TO 1.
SO WE HAVE - SQUARE ROOT 2.
SO THIS SIMPLIFIES TO 2 - SQUARE ROOT 2.
THE OTHER WAY TO SIMPLIFY THIS CORRECTLY
WOULD BE TO TAKE THIS FORM HERE
AND FACTOR OUT THE GREATEST COMMON FACTOR
FROM THE NUMERATOR.
THE GREATEST COMMON FACTOR WOULD BE 4
SO WE COULD WRITE THIS AS 4 x THE QUANTITY 2 - SQUARE ROOT 2
AND THEN DIVIDE THIS BY 4
AND IN THIS FORM SINCE THIS 4 IS ATTACHED BY MULTIPLICATION
WE CAN SIMPLIFY THIS 4 AND THIS 4
LEAVING US WITH 2 - SQUARE ROOT 2.
NOTICE HOW THE RESULT IS THE SAME.
I PREFER THE FIRST METHOD OF SIMPLIFYING
BY WRITING THIS AS TWO FRACTIONS
BUT THIS METHOD IS ALSO VALID.
BUT SOMETIMES THE RESULTS MAY BE IN A DIFFERENT FORM
WHICH WE'LL SEE IN THE SECOND EXAMPLE.
SO AGAIN, THE FIRST STEP HERE IS TO SIMPLIFY
THE SQUARE ROOT OF 54.
WELL 54 = 9 x 6.
9s A PERFECT SQUARE.
THE SQUARE ROOT OF 9 IS 3
AND THE SQUARE ROOT OF 6 DOES NOT SIMPLIFY.
OR AGAIN, IF WE NEED TO
WE CAN WRITE OUT THE PRIME FACTORIZATION OF 54.
WE WOULD HAVE ONE FACTOR OF 2 AND THREE FACTORS OF 3.
HERE'S A PERFECT SCORE FACTOR
WHICH AGAIN WOULD SIMPLIFY TO 3 SQUARE ROOT 6.
SO EITHER WAY WE CAN WRITE THIS AS 12 - 3 SQUARE ROOT 6
DIVIDED BY 6.
AND AGAIN, WE CANNOT JUST SIMPLIFY THE 12 AND THE 6 HERE
BECAUSE WE CANNOT SIMPLIFY ACROSS ADDITION OR SUBTRACTION.
SO ONE METHOD IS TO BREAK THIS UP INTO 2 FRACTIONS
12 DIVIDED BY 6 - 3 SQUARE ROOT 6, DIVIDED BY 6
AND 12 DIVIDED BY 6 IS EQUAL TO 2.
HERE THE 3 SIMPLIFIES TO 1, THE 6 SIMPLIFIES TO 2.
SO THIS SIMPLIFIES TO - SQUARE ROOT 6 DIVIDED BY 2.
NOW LET'S TAKE A LOOK AT THE SECOND METHOD
AND THIS WILL GIVE US A DIFFERENT FORM
OF THE SIMPLIFIED EXPRESSION.
AGAIN LOOKING AT JUST THE NUMERATOR
NOTICE HOW THERE'S A COMMON FACTOR OF 3
SO WE CAN WRITE THIS AS 3 x THE QUANTITY 4 - SQUARE ROOT 6
DIVIDED BY 6 WHICH WE CAN WRITE AS 3 x 2.
IN THIS FORM, WE CAN SEE THAT THERE'S A 3/3
THAT SIMPLIFIES TO 1
LEAVING US WITH 4 - SQUARE ROOT 6 DIVIDED BY 2.
SO THIS IS CONSIDERED SIMPLIFIED
AND NOTICE HOW IT DOESN'T LOOK QUITE THE SAME
AS THE EXPRESSION IN BLUE.
SO THIS IS ACCEPTABLE
BUT JUST TO VERIFY THAT IT IS THE SAME
WE HAVE TO BREAK THIS UP INTO 2 FRACTIONS
WHERE WE HAVE 4/2 - SQUARED ROOT 6/2
NOTICE HOW THIS FIRST FRACTION DOES SIMPLIFY TO 2.
SO THIS IS WHY SOMETIMES THERE CAN BE A LITTLE BIT OF CONFUSION
WHEN SIMPLIFYING EXPRESSIONS LIKE THIS.
THESE TWO FORMS ARE CONSIDERED SIMPLIFIED
EVEN THOUGH THEY DO LOOK A LITTLE BIT DIFFERENT.
SO KEEP THIS IN MIND WHEN APPLYING THE QUADRATIC FORMULA.