(THEME MUSIC)
LET'S TALK EXPONENTS --
THOSE LITTLE NUMBERS
THAT SIT ON THE SHOULDERS OF
OTHER NUMBERS IN EXPRESSIONS.
IN THIS MODULE
WE'LL SEE HOW TO WORK
WITH SYMBOLS LIKE THESE,
WHICH WE CALL
EXPONENTS OR POWERS.
TAKE THIS ONE
FOR EXAMPLE:
THEY ALL MEAN THE
SAME THING --
TAKE THREE COPIES OF "2" AND
MULTIPLY THEM ALL TOGETHER.
2 TIMES 2 TIMES 2,
WHICH EQUALS EIGHT.
2 TIMES 2 IS FOUR,
TIMES 2 IS EIGHT.
DO YOU THINK THAT 2
TO THE THIRD POWER
AND 2 TIMES 2 TIMES 2
AND 8
ARE THREE DIFFERENT WAYS OF WRITING THE SAME THING?
WELL, YOU'RE RIGHT.
AND NOW YOU KNOW A KEY
FACT ABOUT EXPONENTS.
EVERYTHING ELSE
BUILDS ON THAT IDEA.
HERE'S ANOTHER EXAMPLE.
WHAT DOES THIS MEAN?
FIRST OF ALL,
HOW DO WE SAY IT?
HOW ABOUT 3 TO
THE POWER 5
OR 3 TO THE 5TH POWER.
OR 3 TO THE
FIFTH, FOR SHORT?
BUT WHAT DOES THAT MEAN?
HOW ELSE CAN WE WRITE IT?
3 TO THE POWER 5
IS THE SAME AS FIVE COPIES
OF "3" MULTIPLIED TOGETHER.
THAT IS, 3 TIMES 3 TIMES 3
TIMES 3 TIMES 3.
AND IF YOU'RE QUICK
WITH MULIPLICATION,
YOU KNOW ALREADY THAT
THE ANSWER IS 243.
3 TIMES 3 IS 9, TIMES 3
IS 27, TIMES 3 IS 81,
TIMES 3 IS 243.
3 DIFFERENT WAYS OF WRITING
EXACTLY THE SAME THING.
GOT IT? GOOD.
IT'S A KEY CONCEPT.
SO WHAT DO YOU
DO WITH THIS ONE?
X TO THE POWER OF 4
OR X TO THE FOURTH
MEANS FOUR COPIES OF
X MULTIPLIED TOGETHER.
IN THIS CASE, THAT'S
AS FAR AS WE CAN GO
UNLESS WE KNOW
THE VALUE OF "X".
TERMINOLOGY CAN
BE CONFUSING HERE.
SO LET ME POINT
OUT TWO THINGS:
FIRST, WE'VE ALREADY
USED DIFFERENT NAMES
FOR EXPRESSIONS LIKE THESE --
ALL MEANING THE SAME THING.
X TO THE POWER OF FOUR.
X TO THE POWER FOUR,
WITHOUT THE "OF".
X THE FOURTH POWER, OR
SIMPLY X TO THE FOURTH.
IN CASE YOU WEREN'T
CONFUSED ENOUGH,
X TO THE POWER 2 AND
X TO THE POWER 3
HAVE SPECIAL
ADDITIONAL NAMES.
X TO THE SECOND POWER
IS CALLED X SQUARED.
4 TO THE THIRD POWER
IS CALLED 4 CUBED.
ANYTHING TO THE
POWER 2 IS SQUARED.
ANYTHING TO THE
THIRD POWER IS CUBED.
SECOND, WE ALSO NEED NAMES
TO DISTINGUISH THE TWO PARTS
OF EXPRESSIONS LIKE THESE.
SO WE CALL THE BOTTOM
PART THE "BASE".
Z IS A BASE. 8 IS A BASE.
AND WE CALL THE TOP
PART THE EXPONENT.
NOTICE THAT THE
EXPONENT MAY BE WRITTEN
QUITE A BIT SMALLER
THAN THE BASE IS.
NOW WE KNOW WHAT
EXPONENTS ARE.
BUT WHAT CAN WE
DO WITH THEM?
THE ANSWER? PLENTY!
WE CAN DO TRICKS
THAT SEEM LIKE MAGIC.
FOR EXAMPLE, HERE
IS A HARD QUESTION
WITH A VERY EASY ANSWER --
LIKE MAGIC.
HOW DO WE MULTIPLY TWO
NUMBERS WITH THE SAME BASE
WHEN BOTH HAVE EXPONENTS?
LET'S SAY, 2 CUBED
TIMES 2 TO THE POWER 4.
YOU ALREADY KNOW ONE
WAY TO ANSWER THAT.
YOU JUST WRITE OUT 2 CUBED
AS 2 TIMES 2 TIMES 2,
AND 2 TO THE FOURTH AS
2 TIMES 2 TIMES 2 TIMES 2,
AND MULTIPLY THEM ALL
TOGETHER TO FIND THE ANSWER.
HOW ELSE CAN WE WRITE
ALL THESE 2 TIMES 2'S?
IT IS EXACTLY THE
SAME AS WRITING OUT
ALL THE 2 TIMES 2'S
FOR 2 TO THE SEVENTH.
SO HERE IS THE TRICK.
YOU DON'T HAVE
TO GO THROUGH
ALL THAT WRITING OUT
AND COUNTING 2'S.
JUST ADD THE EXPONENTS.
3 PLUS 4 IS 7.
2 TO THE THIRD
TIMES 2 TO THE FOURTH
IS 2 TO THE SEVENTH.
REMEMBER, WE MUST
HAVE EXPRESSIONS
WHERE THE BASE
IS THE SAME.
AND NOTICE THIS: WE ARE
NOT DOING ADDITION HERE.
WHAT WE ARE DOING
IS MULTIPLICATION:
2 TO THE THIRD
TIMES 2 TO THE FOURTH.
THE WAY WE DO
THE MULTIPLYING
IS BY ADDING THE
EXPONENTS --
LIKE MAGIC.
AND THIS SUGGESTS
A SIMPLE RULE
FOR MULTIPLYING POWERS
WITH THE SAME BASE:
ADD THE EXPONENTS.
NOW TRY OUR RULE
ON THIS ONE.
JUST ASK YOURSELF:
"WHAT'S THE RULE?"
"HOW DO I APPLY IT?"
IS THE BASE THE SAME?
IT IS: SEVEN.
SO TO MULTIPLY,
YOU ADD THE EXPONENTS.
5 PLUS 4 IS 9, OR SEVEN
TO THE POWER NINE.
AND YOU DIDN'T
HAVE TO WRITE OUT
7 TIMES 7 TIMES 7
NINE TIMES TO DO IT.
NOW LET'S LOOK AT A TERM
THAT LOOKS EVEN MESSIER:
RAISING ONE POWER
TO ANOTHER POWER.
TAKE 3 SQUARED FOR
EXAMPLE, AND RAISE IT
(THE WHOLE EXPRESSION, 3
SQUARED) TO THE FOURTH POWER.
CAN WE WORK MAGIC,
OR IS IT REALLY A MESS?
WELL, LET'S TRY IT!
DON'T LET IT THROW YOU!
YOU REALLY KNOW HOW
TO DO IT ALREADY.
DO THE OUTSIDE EXPONENT
FIRST - THE POWER OF FOUR.
THAT MEANS WE HAVE 4
OCCURENCES OF 3 SQUARED,
AND THAT THEY ARE ALL
MULTIPLIED TOGETHER.
THEN DO THE
SQUARED PART.
WHEREVER YOU SEE
3 TO THE POWER 2,
JUST WRITE TWO 3'S
MULTIPLIED TOGETHER.
AND YOU FIND THE SORT OF
ANSWER WE SAW BEFORE -
A WHOLE BUNCH OF 3'S
MULTIPLIED TOGETHER.
COUNT THEM ALL, AND THEY COME
OUT 3 TO THE POWER OF 8.
BUT IS THERE A
MAGIC SHORT CUT?
YOU DON'T WANT TO WRITE IT
OUT THE LONG WAY EVERY TIME.
AND YOU DON'T HAVE TO,
BECAUSE YOU GET EXACTLY
THE SAME ANSWER
BY SIMPLY MULTIPLYING
THE TWO EXPONENTS.
THE POWER 2 RAISED
TO THE POWER 4:
MULTIPLY THE EXPONENT 2 BY
THE EXPONENT 4 TO GET 8,
AND YOU HAVE 3
WITH THE EXPONENT 8
OR 3 TO THE 8TH POWER.
AND THAT SUGGESTS
ANOTHER GENERAL RULE:
A RULE FOR RAISING ONE
POWER TO ANOTHER POWER.
YOU GUESSED IT!
MULTIPLY THE EXPONENTS.
OKAY. PAUSE THE PROGRAM
AND GIVE THIS ONE
A TRY ON YOUR OWN.
WHEN YOU HAVE AN ANSWER,
CONTINUE THE SHOW
AND WE WILL COMPARE NOTES.
HOW DID YOU DO?
DID YOU USE OUR
GENERAL RULE?
TO RAISE ONE POWER,
THE POWER 9,
TO ANOTHER, THE POWER 8, MULTIPLY THE EXPONENTS.
9 X 8 IS 72. THE ANSWER
IS 7 TO THE POWER 72.
NOTE TWO THINGS HERE:
FIRST, THESE THREE
EXPRESSIONS ARE IDENTICAL;
THEY ARE DIFFERENT WAYS OF
SAYING EXACTLY THE SAME THING.
SECOND, YOU DIDN'T HAVE
TO WRITE OUT ALL 72 7'S
AND THEN COUNT THEM
TO FIND THE ANSWER.
WITH OUR RULE, WE AVOIDED
THE MESS -- LIKE MAGIC.
SO, IS THIS MAGIC ONLY
FOR MULTIPLICATION?
HOW ABOUT THE OTHER
DIRECTION, DIVISION?
LET'S GO BACK TO WHAT WE FIRST
LEARNED ABOUT MULTIPLICATION
AND FIND OUT.
OUR RULE WAS THIS:
TO MULTIPLY POWERS WITH THE
SAME BASE, WHAT DID WE DO?
WE ADDED THE POWERS,
ADDED THE EXPONENTS.
NOW LET'S SEE WHAT WE
CAN LEARN ABOUT DIVISION.
ONCE AGAIN, 2 EXPRESSIONS
WITH THE SAME BASE.
BOTH HAVE EXPONENTS.
HOW DO WE DIVIDE
5 TO THE POWER 7
BY 5 TO THE POWER 3?
FIRST, AS ALWAYS, WE
DO IT THE EASY WAY,
WHICH ALWAYS GETS MESSY AND
IS REALLY THE HARD WAY.
WE HAVE SEVEN OCCURENCES
OF 5 MULTIPLIED TOGETHER
AND WE DIVIDE THAT
BY THREE OCCURENCES OF 5 MULTIPLIED TOGETHER,
GIVING US THIS ODD
LOOKING FRACTION.
CAN WE REDUCE THE
FRACTION BY CANCELLING?
WE CAN.
WE DIVIDE BOTH TOP AND BOTTOM
BY 5, THREE TIMES OVER,
WHICH LEAVES US WITH
FOUR OCCURENCES OF 5
MULTIPLIED TOGETHER,
OR 5 TO THE POWER 4.
SO OUR DIVISION THE EASY
WAY WAS A LOT OF WORK.
IS THERE A
SHORTCUT THERE?
WITHOUT ALL THAT COUNTING
AND CANCELLING OF 5'S?
THERE IS.
AND IT'S JUST THE OPPOSITE
OF WHAT WE DID TO MULTIPLY.
SO WHAT DO YOU THINK
THE RULE WOULD BE?
TO DIVIDE POWERS WITH THE
SAME BASE, YOU DO WHAT?
YOU'VE GOT IT! YOU SUBTRACT EXPONENTS, TOP MINUS BOTTOM.
BUT SOME EXPRESSIONS
COULD BE A PROBLEM.
WHAT IF THE HIGHER POWER
(THE LARGER EXPONENT)
IS ON THE BOTTOM?
APPLY THE RULE
AND WHAT DO WE GET?
A HEADACHE, BECAUSE 3 MINUS 5
IS A NEGATIVE 2.
8 TO THE POWER
OF A NEGATIVE 2?
WHAT IN THE WORLD
DOES THAT MEAN?
WE CAN'T LET 8 HAPPEN
NEGATIVE 2 TIMES
AND THEN MULTIPLY
THEM TOGETHER,
SO WE'D BETTER FIND
OUT WHAT IT DOES MEAN.
LET'S GO BACK TO DOING THIS PROBLEM THE EASY HARD WAY.
WE ALREADY KNOW THAT 8 TO
THE 3RD OVER 8 TO THE 5TH
CAN BE WRITTEN LIKE THIS.
AND WE KNOW WE CAN
REDUCE THE FRACTION
BY CANCELLING OUT
THREE OF THE 8'S.
1 OVER 8 TO THE POWER 2.
THIS ANSWER IS CORRECT.
THE ANSWER WE GOT BY FOLLOWING
OUR SUBTRACTION RULE,
8 TO THE POWER OF NEGATIVE
2, WAS ALSO CORRECT,
WHICH MEANS THE TWO
ARE DIFFERENT WAYS
OF SAYING THE
SAME THING.
AND SINCE ALL NEGATIVE
EXPONENTS ACT THIS WAY,
WE CAN MAKE A RULE
FOR ALL OF THEM.
HERE IS THE EASIEST
WAY TO PUT IT:
THE NEGATIVE SIGN IN A NEGATIVE
EXPONENT MEANS 1 OVER,
GIVING US A FRACTION.
JUST PICTURE THAT MINUS SIGN
TURNING THE EXPRESSION INTO
ONE OVER THE BASE
TO A POSITIVE POWER,
AND YOU'VE GOT IT.
AND THAT'S GREAT,
BECAUSE IT MEANS OUR RULE
CAN WORK FOR ALL EXPONENTS,
POSITIVE OR NEGATIVE.
LET'S TRY THIS EXAMPLE.
OUR DIVISION RULE REQUIRES
THE SUBTRACTION OF EXPONENTS.
IN THIS CASE, THE SUBTRACTION
OF NEGATIVE NUMBERS,
WHICH YOU KNOW HOW
TO DO, RIGHT?
OKAY, USING THE
DIVISION RULE,
WE SUBTRACT EXPONENTS -
TOP MINUS BOTTOM -
WHICH GIVES YOU A
SIMPLE PROBLEM
IN THE SUBTRACTION OF
NEGATIVE NUMBERS
WHERE MINUS A NEGATIVE,
OR A DOUBLE NEGATIVE,
EQUALS A POSITIVE OR PLUS.
AND THE ANSWER IS SIMPLY NEGATIVE 3 PLUS 5.
OR POSITIVE 2.
X TO THE POWER 2.
X SQUARED.
YOU MAY THINK WE HAVE A LOT
TO REMEMBER ABOUT EXPONENTS,
BUT IT ALL COMES DOWN TO
THESE 3 SIMPLE STATEMENTS.
IF YOU REMEMBER THESE RULES,
WE ARE READY TO MOVE ON.
IF YOU DON'T,
YOU MIGHT WANT TO REVIEW
THE TROUBLESOME ONES.
OKAY, BEFORE WE FINISH LET'S
LOOK AT TWO SPECIAL CASES.
THE FIRST IS VERY SIMPLE.
WHAT DOES X TO THE
POWER ZERO MEAN?
DON'T PANIC! JUST USE
WHAT YOU ALREADY KNOW.
LET'S LOOK AT AN EXAMPLE WHERE
WE CAN USE OUR FIRST RULE:
X TO THE POWER 0
MULTIPLIED BY X
TO THE POWER 3.
WHICH, ODDLY, GIVES
YOU X TO THE POWER 3!
ONE OF THE SAME THINGS
YOU STARTED WITH.
AND THAT HAS TO MEAN
X TO THE POWER ZERO
IS THE NUMBER 1.
SO AS LONG AS WE NEVER
USE ZERO AS A BASE,
WHICH WOULD GET
US NOWHERE FAST,
ANY BASE TO THE
POWER ZERO EQUALS 1!
AND THAT LEAVES ONLY ONE
MORE SPECIAL EXPONENT
TO TALK ABOUT:
X TO THE POWER 1.
WHAT CAN WE SAY ABOUT IT?
X TO THE POWER 1 IS X
ONE TIME, OR JUST X!
BUT YOU KNEW THAT, RIGHT?
WE NEED TO ADD ONE
FINAL RULE TO OUR LIST.
QUESTION: WHAT
IS ANOTHER WAY
OF WRITING THIS EXPRESSION?
YOU CAN WRITE IT LIKE THIS.
IF YOU WANT TO SEE
WHY THAT WORKS,
PICK A NUMBER FOR "N" AND
SOLVE IT THE HARD EASY WAY,
LIKE WE DID FOR
ALL THE OTHER RULES.
NOTICE THAT
THIS IS A CASE
OF THE SAME EXPONENT
AND DIFFERENT BASES.
THAT'S JUST THE OPPOSITE OF
WHAT WE WERE LOOKING AT BEFORE.
AND THAT DOES IT. CONGRATULATIONS.
YOU HAVE MASTERED
ALL THE RULES
THAT TELL US HOW TO
MULTIPLY AND DIVIDE
EXPRESSIONS WITH EXPONENTS.
NOTICE THAT WE DO
NOT HAVE A RULE YET
FOR ADDING OR SUBTRACTING
EXPRESSIONS WITH EXPONENTS.
YOUR BOOK HAS LOTS
MORE EXAMPLES
OF EVERYTHING WE
HAVE SEEN HERE.
TO PRACTICE WHAT
YOU'VE LEARNED,
LET'S TAKE A SWING AT THIS
COMPLICATED LOOKING THING.
FIRST, LET ME ASURE YOU:
YOU DO KNOW HOW TO DO IT.
DON'T PANIC.
USE WHAT YOU KNOW.
TAKE ONE LETTER AT A TIME,
AND GO FOR IT!
AND IF I WERE YOU, I'D
REDUCE THOSE NUMBERS FIRST.
PAUSE THE PROGRAM
WHILE YOU WORK.
WHEN YOU HAVE IT SOLVED,
CLICK PLAY AND WE
WILL COMPARE NOTES.
WE CAN SOLVE THIS
EXPRESSION IF WE KEEP COOL
AND USE WHAT WE KNOW,
STEP BY STEP.
WE CAN CERTAINLY
REDUCE THOSE NUMBERS
BY DIVIDING THE TOP
AND BOTTOM BY 7,
WITHOUT EFFECTING
THE REST IN ANY WAY.
THAT LOOKS LESS
MESSY ALREADY.
THEN WE CAN TAKE THAT XY
TO THE 3RD ON THE TOP
AND REWRITE IT.
X TO THE 3RD,
Y TO THE 3RD,
AND IT STILL MEANS
THE SAME THING.
THAT LET'S US USE THE
RULE FOR DIVISION.
REMEMBER? SUBTRACT
EXPONENTS, TOP MINUS BOTTOM.
SUBTRACT THE EXPONENTS
FOR THE "X'S" FIRST
AND THEN THE "Y'S".
WE DO THEM AS TWO
SEPARATE PROBLEMS
AND END UP WITH AN EXPONENT
OF 4 FOR X AND 7 FOR Y.
SO OUR FINAL ANSWER IS
5X TO THE 4TH,
Y TO THE 7TH.
DO YOU SEE HOW BIG
MESSY EXPRESSIONS
CAN BE EASY TO SIMPLIFY
WHEN YOU TAKE THEM ONE
SMALL STEP AT A TIME?
BE SURE TO TRY MORE
PROBLEMS RIGHT AWAY
WHILE THE EXPONENT RULES
ARE STILL FRESH IN YOUR MIND.
FIND THEM IN YOUR
TEXTBOOK OR STUDY GUIDE.
THESE FOUR STATEMENTS TELL YOU
EVERYTHING YOU NEED TO KNOW
TO GET THE
ANSWERS YOU WANT.
[THEME MUSIC]
CAPTIONS PROVIDED BY
THE DISABILITY INSTRUCTIONAL SUPPORT CENTER