- 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.
