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- WE WANT TO SIMPLIFY EACH EXPRESSION COMPLETELY
USING OUR EXPONENT RULES GIVEN HERE ON THE RIGHT.
IF WE TAKE A LOOK AT THIS FIRST EXPRESSION,
THE FIRST THING I WOULD PROBABLY DO
IS PUT AN EXPONENT OF 1 ON THIS S.
NOTICE HOW WE HAVE THIS FRACTION HERE RAISED TO THE POWER OF 3.
SO WE ACTUALLY HAVE A CHOICE OF HOW WE WANT TO APPROACH THIS.
IF WE RECOGNIZE THIS INNER FRACTION IS GOING TO SIMPLIFY,
WE COULD SIMPLIFY THIS FIRST, AND THEN ONCE IT'S SIMPLIFIED
WE COULD RAISE IT TO THE THIRD POWER.
OR WE COULD APPLY THE POWER TO POWER PROPERTY AT FIRST
AND THEN SIMPLIFY.
BUT FOLLOWING THE ORDER OF OPERATIONS,
LET'S GO AHEAD AND SIMPLIFY INSIDE THE PARENTHESES FIRST.
NOTICE WE HAVE S TO THE FOURTH OVER S TO THE FIRST,
AND THEN WE HAVE T TO THE FIFTH OVER T TO THE THIRD.
SO BECAUSE WE HAVE A QUOTIENT,
OR WE'RE DIVIDING,
AND THE BASES ARE THE SAME, WE SUBTRACT OUR EXPONENTS.
SO FOR S TO THE FOURTH DIVIDED BY S TO THE FIRST,
WE WOULD HAVE S TO THE POWER OF 4 - 1
OR S TO THE THIRD.
AND THEN FOR T TO THE FIFTH DIVIDED BY T TO THE THIRD,
WE'D HAVE T TO THE POWER OF 5 - 3,
WHICH WOULD BE T TO THE SECOND.
AND AGAIN, ALL OF THIS IS RAISED TO THE THIRD POWER,
AND NOW WE SHOULD RECOGNIZE THAT WE HAVE
POWERS RAISED TO POWERS, WHICH IS THIS RULE HERE.
SO WHEN WE HAVE A POWER RAISED TO A POWER
WE MULTIPLY THE EXPONENTS.
SO HERE WE'D HAVE S TO THE POWER OF 3 x 3,
THAT WOULD BE 9, AND T TO THE POWER OF 2 x 3,
WHICH WOULD BE T TO THE SIXTH.
AND AGAIN, IF YOU FIND YOURSELF BEING OVERWHELMED
BY ALL THESE RULES, AS LONG AS YOU KNOW
WHAT AN EXPONENT MEANS, WE COULD ALSO GET
THE SAME RESULT.
WHAT I MEAN BY THAT IS IF WE TAKE A LOOK
AT THIS INNER FRACTION, AND WE KNOW S TO THE FOURTH
WOULD BE 4 FACTORS OF S AND T TO THE FIFTH
WOULD BE 5 FACTORS OF T.
S TO THE FIRST WOULD BE 1 FACTOR OF S,
AND THEN WE HAVE 3 FACTORS OF T.
WE KNOW ANY TIME WE HAVE A FACTOR OVER ITSELF
IT SIMPLIFIES TO 1, SO THIS SIMPLIFIES TO 1.
AND THEN WE HAVE 3 FACTORS OF T
THAT SIMPLIFY TO 1.
SO NOW LOOKING AT THE INNERMOST FRACTION,
NOTICE HOW WE HAVE S TO THE THIRD,
T TO THE SECOND RAISED TO THE THIRD,
WHICH IS WHAT WE HAVE HERE,
AND THEN WE COULD ALSO EXPAND THIS AGAIN.
WE KNOW SOMETHING RAISED TO THE THIRD POWER MEANS
WE'LL HAVE 3 FACTORS OF S TO THE THIRD
T TO THE SECOND,
AND THEN MAYBE FROM HERE WE'D BE OKAY.
NOTICE HOW WE HAVE 3, 6, 9 FACTORS OF S
AND 2, 4, 6 FACTORS OF T.
SO THE RESULT, OF COURSE, WOULD BE THE SAME,
BUT IT WOULD TAKE A LITTLE BIT MORE WORK.
LOOKING AT THE SECOND EXAMPLE, IT MAY BE TEMPTING
TO TRY TO SIMPLIFY OUT THESE TWOS
AND MAYBE EVEN SOME OF THESE Xs, BUT WE CAN'T DO THAT,
BECAUSE NOTICE HOW THE NUMERATOR
IS BEING RAISED TO THE FOURTH POWER.
AND AGAIN, IT'S IMPORTANT TO RECOGNIZE
THAT THE POWER ON THE 2 IS 1.
SO LOOKING AT THE NUMERATOR WE HAVE POWERS TO POWERS,
SO WE'RE GOING TO MULTIPLY OUR EXPONENTS.
WE WOULD HAVE 2 TO THE POWER OF 1 x 4,
THAT'S 4, AND THEN X TO THE POWER OF 4 x 4,
THAT'S 16, ALL OVER 2X TO THE FIFTH.
AGAIN, THIS WOULD BE 2 TO THE FIRST.
SO WE'D HAVE 2 TO THE POWER OF 4 - 1,
THAT'S 3, AND THEN X TO THE POWER OF 16 - 5,
THAT WOULD BE 11.
THE LAST THING WE CAN DO IS EVALUATE 2 TO THE THIRD.
THAT WOULD BE 2 x 2 x 2, WHICH IS EQUAL TO 8.
SO WE HAVE 8X TO THE ELEVENTH.
THANK YOU FOR WATCHING.