- WE WANT TO SIMPLIFY THE FOLLOWING EXPRESSIONS.
IF WE NEED THE RULES FOR EXPONENTS,
THEY'RE PROVIDED HERE IN RED.
LOOKING AT OUR FIRST EXAMPLE, WE HAVE 2X TO THE FIFTH
RAISED TO THE 0 POWER.
WELL, THIS ONE SHOULD BE PRETTY STRAIGHTFORWARD.
LOOKING AT OUR EXPONENT RULES, "A" TO THE ZERO IS EQUAL TO 1,
SO ANYTHING RAISED TO THE 0 POWER
IS ALWAYS EQUAL TO 1.
LOOKING AT THE SECOND EXAMPLE, WE HAVE -3X TO THE -8 x 4X
TO THE FIFTH.
WELL, -3X TO THE -8 MEANS -3 x X TO THE -8,
AND 4X TO THE FIFTH MEANS 4 x X TO THE FIFTH.
AND WE KNOW WE CAN CHANGE THE ORDER OF MULTIPLICATION
AND NOT CHANGE THEIR PRODUCT, SO IF IT'S HELPFUL,
WE CAN WRITE THIS AS -3 x 4 x X TO THE -8 x X
TO THE FIFTH.
THIS IS CALLED THE COMMUTATIVE PROPERTY
IN MULTIPLICATION.
AND NOW IT'S A LITTLE BIT EASIER TO SEE THAT WE HAVE -12,
AND THEN FOR X TO THE -8 x X
TO THE FIFTH, SINCE THE BASES ARE THE SAME
AND WE'RE MULTIPLYING, WE ADD THE EXPONENTS.
-8 + 5 = -3, SO WE HAVE X TO THE -3.
BUT WE CAN'T LEAVE IT LIKE THIS, BECAUSE WE'RE NOT ALLOWED
TO HAVE NEGATIVE EXPONENTS IN A SIMPLIFIED EXPRESSION.
SO NOW WE USE THIS RULE HERE THAT SAYS
"A" TO THE POWER OF -M = 1 OVER "A" TO THE POWER OF M.
IN THE MORE GENERAL SENSE, IF WE MAKE THIS A FRACTION
BY PUTTING IT OVER 1, IF AN EXPONENTIAL EXPRESSION
CROSSES THE FRACTION BAR, IT'S GOING TO CHANGE
THE SIGN OF THE EXPONENT.
SO IF WE THINK OF THIS AS A FRACTION OVER 1,
WE CAN THEN MOVE X TO THE -3 TO THE DENOMINATOR,
AND IT WILL BECOME X TO THE 3.
SO THIS SIMPLIFIES TO -12 ALL OVER X TO THE POWER OF 3.
LOOKING AT THE NEXT EXAMPLE, WE CAN'T FIND THIS PRODUCT
UNTIL WE FIRST SQUARE 5Y TO THE FOURTH.
AND AGAIN, WE CAN THINK OF THIS AS 5 TO THE FIRST,
Y TO THE FOURTH.
AND WE HAVE POWERS TO POWERS, SO WE'RE GOING TO MULTIPLY
THE EXPONENTS.
SO WE'LL HAVE 5 TO THE SECOND POWER,
Y TO THE 4 x 2, WHICH WOULD BE EIGHTH POWER,
x Y TO THE TENTH.
AND WHILE 5 SQUARED WOULD BE 25, AND THEN
Y TO THE EIGHTH x Y TO THE TENTH,
WE'RE MULTIPLYING.
THE BASES ARE THE SAME, SO WE'RE GOING TO ADD
THE EXPONENTS, SO WE'LL HAVE Y TO THE POWER OF 18.
LOOKING AT OUR LAST EXAMPLE, WE'LL SIMPLIFY THIS
IN THREE PARTS.
WE'LL FIRST DEAL WITH THE QUOTIENT INVOLVING BASE A,
THEN BASE B, AND THEN BASE C, THOUGH IT WILL
PROBABLY BE HELPFUL TO RECOGNIZE THAT THE C
IS GOING TO HAVE AN EXPONENT OF 1.
AND SINCE WE HAVE THREE QUOTIENTS HERE
INVOLVING THE SAME BASE, WE WILL BE SUBTRACTING
THE EXPONENTS.
NOTICE HOW IT'S ALWAYS GOING TO BE THE EXPONENT
OF THE NUMERATOR MINUS THE EXPONENT
OF THE DENOMINATOR.
SO WE'RE GOING TO HAVE "A" TO THE POWER
OF -7 - 3 = -10, B TO THE POWER OF 6 - -2 = 6 + 2 = 8,
AND THEN C TO THE POWER OF 1 - 3 = -2.
SO AGAIN, IF WE THINK OF THIS AS A FRACTION
WITH A DENOMINATOR OF 1, WE CAN CHANGE THE SIGN
OF THIS EXPONENT HERE AND HERE BY MOVING THEM
TO THE DENOMINATOR.
NOTICE HOW THE B HAS A POSITIVE EXPONENT,
SO IT'S GOING TO STAY IN THE NUMERATOR.
SO WE'LL HAVE B TO THE EIGHTH IN THE NUMERATOR,
AND THE DENOMINATOR IS GOING TO BE
"A" TO THE TENTH AND C TO THE SECOND.
I HOPE THESE EXAMPLES WERE HELPFUL.