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WELCOME TO NEGATIVE EXPONENTS.
THE GOAL OF THIS VIDEO IS TO SIMPLIFY EXPRESSIONS,
INVOLVING NEGATIVE EXPONENTS.
WHEN WORKING WITH NEGATIVE EXPONENTS,
ALL OF THE BASIC EXPONENT PROPERTIES STILL APPLY,
SO WE WILL BE USING ALL OF THESE PROPERTIES
AS WE GO THROUGH OUR EXAMPLES.
IF YOU NEED TO REVIEW THESE,
YOU SHOULD WATCH THE VIDEO
ENTITLED "PROPERTIES OF EXPONENTS."
SO HERE ARE THE PROPERTIES OF NEGATIVE EXPONENTS.
A TO THE POWER OF -N = 1/8 OF THE POWER OF +N.
SO FOR EXAMPLE, IF WE HAVE 3 RAISED TO THE POWER OF -2
THAT WOULD = 1/3 TO THE +2 POWER,
WHICH WOULD BE 1 NINTH.
NOW, FOR ME IT'S OFTEN HELPFUL
TO THINK OF 3 TO THE POWER OF - 2 AS A FRACTION,
WHERE ALL OF THIS IS OVER 1.
SO IF I MOVE THIS ACROSS THE FRACTION BAR,
IT CHANGES THE SIGN OF THE EXPONENT FROM A -2 TO A +2.
IT'S ALSO TRUE THAT A TO THE -N POWER
AND A TO THE +N POWER ARE RECIPROCALS.
REMEMBER IF TWO TERMS ARE RECIPROCALS
THEIR PRODUCT WOULD BE ONE.
AND FROM WHAT WE LEARNED IN THE PREVIOUS VIDEO,
IF WE'RE MULTIPLYING AND THE BASES ARE THE SAME,
WE ADD EXPONENTS.
WELL, -N + N = 0, SO 8 OF THE 0 POWER = 1.
OR FROM THIS DEFINITION WE COULD REWRITE A TO THE POWER OF -N
AS 1/A TO THE POWER OF N x A TO THE N.
AGAIN, THIS COULD BE VIEWED AS OVER ONE,
WHICH SIMPLIFIES NICELY 2 = 1.
SO WHICHEVER WAY YOU LOOK AT IT, THESE TWO ARE RECIPROCALS.
THE NEXT PROPERTY STATES
A TO THE -N DIVIDED BY B TO THE -M = B TO THE +M
DIVIDED BY A TO THE +N.
AND THIS IS KIND OF THE IDEA I WAS ALLUDING TO ABOVE.
IF A FACTOR IS MOVED ACROSS THE FRACTION BAR,
IT WILL CHANGE THE SIGN OF THE EXPONENT.
SO NOTICE HOW THIS IS MOVED UP TO NUMERATOR,
IT BECOMES A +M,
AND IF THIS MOVES DOWN TO THE DENOMINATOR,
IT BECOMES A +N POWER.
LOOKING AT A NUMERICAL EXAMPLE,
IF WE ADD 2 TO THE POWER OF -2 DIVIDED BY 5 TO THE POWER OF -2,
IF WE MOVE THIS UP, IT CHANGES THE SIGN OF THE EXPONENT.
IF WE MOVE THIS DOWN, IT CHANGES THE SIGN OF THE EXPONENT.
SO THIS WOULD EQUAL 5 TO THE +2 POWER OF 5 SQUARED,
OR 2 SQUARED, WHICH WOULD EQUAL 25/4.
AND THE LAST PROPERTY OF NEGATIVE EXPONENTS,
ANY BASE TO AN EXPONENT IS EQUAL TO THE RECIPROCAL OF THE BASE
IF RAISED TO THE OPPOSITE EXPONENT.
SO NOTICE HOW IF WE HAVE A/B RAISED TO THE -N POWER,
THAT IS EQUAL TO THE RECIPROCAL OR B/A RAISED TO THE +N POWER.
SO IF WE HAVE 3/4 RAISED TO THE -2 POWER
THAT IS THE SAME AS 4/3 RAISED TO THE +2 POWER,
WHICH WOULD BE 16/9.
NOW, IN MY MIND, THIS REALLY IS NOT A NEW RULE
BECAUSE WE KNOW THE EXPONENT ON THIS A IS 1,
AND THE EXPONENT ON THE B IS 1.
SO FROM THE PREVIOUS VIDEO,
WE KNOW IF WE APPLY THE POWER RULE HERE
WE WOULD HAVE A TO THE POWER OF -N/B TO THE POWER OF -N,
WHICH REALLY BRINGS US BACK TO THE PREVIOUS RULE
WHERE IF WE MOVE THESE ACROSS THE FRACTION BAR
THAT CHANGES THE SIGN OF THE EXPONENT.
AND IF WE WANTED TO,
WE COULD REWRITE THIS AS D/A RAISED TO THE N POWER.
SO ALL OF THESE RULES ARE CLOSELY RELATED,
BUT I WOULD SAY THE MAIN RULE YOU GET TO BE AWARE OF,
IS IF YOU MOVE SOMETHING ACROSS THE FRACTION BAR,
IT WILL CHANGE THE SIGN OF THE EXPONENT.
SO LET'S GO AHEAD AND APPLY THESE NEW RULES WITH
THE BASIC PROPERTIES OF EXPONENTS.
HERE WE WANT TO SIMPLIFY
AND WRITE OUR ANSWER WITHOUT NEGATIVE EXPONENTS.
SO HERE WE'RE MULTIPLYING, OUR BASES ARE THE SAME.
SO THE RULE IS WE WILL ADD OUR EXPONENTS.
THAT'S THE PRODUCT RULE OVER HERE ON THE RIGHT.
SO WE'D HAVE 2 RAISED TO THE POWER OF -5
+ 2 = 2 TO THE POWER OF -3.
BUT WE CANNOT HAVE NEGATIVE EXPONENTS IN OUR ANSWER.
WE COULD MAKE THIS INTO A FRACTION BY PUTTING OVER ONE.
SO IF WE MOVE THIS DOWN TO THE DENOMINATOR
IT'LL EQUAL 1 DIVIDED 2 TO THE THIRD POWER,
WHICH WOULD = 1/8.
SO THE NEXT EXAMPLE, WE'RE NOW DIVIDING,
SO WE'LL APPLY THE QUOTIENT RULE
WHERE WE SUBTRACT OUR EXPONENTS,
AND IT'S ALWAYS THE EXPONENT OF THE NUMERATOR
MINUS THE EXPONENT OF THE DENOMINATOR.
SO THIS WOULD EQUAL X TO THE POWER OF -4.
BUT, AGAIN, TO HAVE OUR ANSWER WITH A POSITIVE EXPONENT,
WE JUST NEED TO MOVE THIS DOWN TO THE DENOMINATOR,
WHICH WOULD EQUAL 1/X TO THE FOURTH POWER.
ON OUR NEXT EXAMPLE ALL OF THIS IS BEING MULTIPLIED TOGETHER.
SO 6 x 2 THAT WOULD GIVE US 12, M TO THE -4 x M TO THE -2,
AGAIN, WE ADD OUR EXPONENTS TO GET M TO THE -6 POWER.
AND THEN N TO THE SIXTH X N TO THE FIRST = N TO THE SEVENTH.
AND I ALWAYS FIND THIS HELPFUL WHEN I FIND A NEGATIVE EXPONENT
TO MAKE IT INTO A FRACTION.
SO IN ORDER TO HAVE POSITIVE EXPONENT HERE,
- WE'LL MOVE THE BASE OF M DOWN TO THE DENOMINATOR.
SO WE'LL HAVE 12N TO THE SEVENTH AND THE NUMERATOR,
AND M TO THE SIXTH AND THE DENOMINATOR.
HERE WE APPLY THE POWER RULE, WELL, -9 x -4 WOULD BE A +36,
AND THAT WOULD BE OUR FINAL SIMPLIFIED ANSWER.
ON THIS NEXT PROBLEM THERE'S A LOT GOING ON,
AND I ALMOST TAKE A LOOK AT THIS AND SEE FOUR PROBLEMS
WHERE WE HAVE THE CONSTANTS,
WE HAVE THE X's, AND WE HAVE THE Y's, AND WE HAVE THE Z's.
AND THEY'RE ALL QUOTIENTS.
SO WE'LL SIMPLIFY EACH QUOTIENT SEPARATELY
AND THEN COMBINE THEM FOR OUR FINAL ANSWER.
SO - 15 45THS = -1 1/3.
X TO THE -1 DIVIDED BY X TO THE -4,
-1 - -4 WOULD GIVE US X TO THE THIRD.
Y TO THE FOURTH DIVIDED BY Y TO THE FIRST,
4 - 1 WOULD GIVE US Y TO THE THIRD.
AND Z TO THE -3 DIVIDED BY Z TO THE 2,
-3 - 2 WOULD BE Z TO THE -5.
SO THE ONLY ISSUE HERE
IS WE CANNOT HAVE Z TO THE -5 IN OUR SIMPLIFIED ANSWER.
REMEMBER IF THIS WAS IN FRACTION FORM
WE'D HAVE A DENOMINATOR OF ONE.
SO WE NEED TO TAKE THIS BASE
AND MOVE IT TO THE DENOMINATOR.
SO OUR FINAL FRACTION WOULD BE - 1 x X CUBED x Y CUBED,
WHICH IS X CUBED Y CUBED OVER 3 Z TO THE FIFTH POWER.
OKAY. ON THIS NEXT PROBLEM,
EVERYTHING HERE HAS BEEN MULTIPLIED TOGETHER.
SO 6 x 2 = 12, M TO THE -4 x M TO THE -2,
WE'RE MULTIPLYING, SO WE ADD OUR EXPONENT.
THIS WOULD BE M TO THE -6.
THIS IS N TO THE SIXTH X N TO THE FIRST,
THAT WOULD BE N TO THE SEVENTH.
AGAIN, IF THIS IN FRACTION FORM WE'D HAVE A ONE DENOMINATOR.
THEREFORE, TO MAKE THIS A POSITIVE EXPONENT,
WE CAN MOVE IT ACROSS THE FRACTION BAR
OR DOWN TO THE DENOMINATOR,
WHICH WOULD GIVE US 12 N TO THE SEVENTH
DIVIDED BY M TO THE SIXTH.
LET'S GO AND TAKE A LOOK AT TWO MORE.
HERE WE HAVE POWERS RAISED TO POWERS,
SO WE ARE GOING TO MULTIPLY OUR EXPONENTS.
THIS WOULD BE X TO THE -4 x 2, THAT'S -8.
AND Y TO THE -3 x 2 WOULD BE -6 POWER.
SO NOW IF WE MOVE THIS ACROSS A FRACTION BAR
AND MOVE THIS UP SO WE'D HAVE Y TO THE SIXTH,
AND NUMERATOR, AND X TO THE EIGHTH, AND THE DENOMINATOR.
OKAY. WE HAVE ONE MORE.
THERE'S QUITE A BIT GOING ON IN THIS PROBLEM.
NOTICE THAT WE DO HAVE POWERS TO POWERS,
SO WE'LL APPLY THAT RULE FIRST.
SO IN OUR NUMERATOR THIS WOULD BE
A TO THE 2 x 3
OR A TO THE SIXTH x HERE WE HAVE A -14 AND B TO THE SEVENTH.
AND OUR DENOMINATOR, WE'D HAVE A TO THE -1 x 2,
A TO THE -2 POWER, B TO THE +2 POWER.
NOW, WE DO HAVE AN OPTION ON WHAT WE WANT TO DO NEXT.
I WOULD PROBABLY GO AHEAD AND MULTIPLY THESE TWO TOGETHER.
WHERE WE MULTIPLY WE WILL ADD THESE EXPONENTS.
WE WOULD OBTAIN A TO THE -8 POWER,
AND EVERYTHING ELSE REMAINS THE SAME.
SO THIS NEXT STEP,I'LL LOOK AT THIS AS TWO PROBLEMS
WHERE WE WANT TO SIMPLIFY THE A's AND THEN THE B's
BY APPLYING THE QUOTIENT RULE WHERE WE SUBTRACT.
SO HERE WE'D HAVE A TO THE -8 - -2.
THAT WOULD GIVE US A -6.
NEXT WE WOULD HAVE B TO THE 7 - 2, WHICH IS B TO THE FIFTH.
AGAIN, WE'RE NOT ALLOWED
TO HAVE NEGATIVE EXPONENTS IN OUR SIMPLIFIED ANSWER,
SO WE WANT TO MOVE THIS DOWN TO THE DENOMINATOR.
SO WE HAVE B TO THE FIFTH POWER OVER A TO THE SIXTH POWER.
OKAY. I HOPE YOU FOUND THIS VIDEO HELPFUL.
THANK YOU FOR WATCHING.