- WE WANT TO SOLVE THE GIVEN
EXPONENTIAL EQUATIONS
BY USING LIKE BASES.
THIS MEANS WE WANT TO WRITE
BOTH SIDES OF THE EQUATION
WITH THE SAME BASE.
IF WE CAN DO THIS
AND THE BASES ARE THE SAME,
THEN THE EXPONENTS
MUST BE EQUAL TO EACH OTHER.
IN GENERAL, IF A TO THE M IS
= TO A TO THE N, THEN M = M.
AGAIN, IF WE HAVE
TWO EXPONENTIAL EXPRESSIONS
EQUAL TO EACH OTHER
AND THE BASES ARE THE SAME,
THEN THE EXPONENTS
MUST BE EQUAL.
SO LOOKING AT OUR FIRST EXAMPLE,
WE HAVE 27 RAISED TO THE POWER
OF X = 2,187.
WELL, WE CAN'T WRITE 2,187
WITH A BASE OF 27,
SO WHAT WE'LL HAVE TO DO
IS WRITE THE LEFT SIDE
OF THE EQUATION
USING A DIFFERENT BASE.
AND TO FIGURE OUT
HOW TO DO THIS,
IT'S OFTEN HELPFUL TO TAKE A
LOOK AT THE PRIME FACTORIZATION.
SO 27 IS = TO 9 x 3
AND 9 IS = TO 3 x 3.
SO 27 IS =
TO THREE FACTORS OF 3,
WHICH MEANS WE CAN WRITE
THE LEFT SIDE OF THE EQUATION AS
3 TO THE THIRD
RAISED TO THE POWER OF X =
NOW, WE WANT TO SEE IF WE CAN
WRITE 2,187 WITH A BASE OF 3,
WHICH IS NOT AN EASY TASK.
REMEMBER SINCE THE SUM
OF THE DIGITS IS DIVISIBLE BY 3,
SO IS THIS NUMBER.
MEANING SINCE 2 + 1 + 8 + 7
IS = TO 18,
WHICH IS DIVISIBLE BY 3,
SO IS THIS NUMBER.
SO WE KNOW IT'S GOING TO BE
3 x SOMETHING.
IF WE NEEDED TO,
WE COULD USE THE CALCULATOR
AND TAKE 2,187
AND DIVIDE IT BY 3 TO HELP US,
IT'S = TO 729.
729's ALSO DIVISIBLE BY 3.
AGAIN, IF WE NEED TO WE CAN USE
THE CALCULATOR AGAIN,
729 DIVIDED BY 3 IS = TO 243.
2 + 4 + 6 = 9,
WHICH IS DIVISIBLE BY 3,
SO IS THIS NUMBER.
THIS IS 3 x 81,
81 IS = TO 9 x 9,
AND 9 IS = TO 3 x 3.
SO WE HAVE 1, 2, 3, 4, 5, 6 7
FACTORS OF 3.
SO WE CAN WRITE THE RIGHT SIDE
AS 3 RAISED TO POWER OF 7.
NOW, WE DO HAVE A COMMON BASE,
BUT NOTICE ON THE LEFT SIDE WE
HAVE A POWER RAISED TO A POWER,
AND THE RULE IS THAT WE CAN
MULTIPLY THESE EXPONENTS.
SO WE CAN WRITE THIS AS 3
RAISED TO THE POWER OF 3X
= 3 TO THE SEVENTH.
AND THIS FORM,
SINCE THESE ARE EQUAL
AND THE BASES ARE THE SAME,
WE KNOW THAT 3X MUST = 7.
AND THIS WILL ALLOW US
TO SOLVE FOR X.
IF 3X = 7,
WE'LL DIVIDE BOTH SIDES BY 3,
AND WE HAVE X = 7/3.
LET'S TAKE A LOOK
AT OUR SECOND EXAMPLE.
HERE WE HAVE 1/16 RAISED
TO THE POWER OF X IS = TO 32.
AGAIN, OUR GOAL IS TO WRITE
THE LEFT SIDE
AND THE RIGHT SIDE OF THE
EQUATION WITH THE SAME BASE.
WELL, I KNOW 16 IS
= TO 4 x 4 AND 4 IS = TO 2 x 2.
SO WE COULD WRITE THIS AS
1/2 TO THE FOURTH
RAISED TO THE POWER OF X =--
EVEN THOUGH THIS IS
IN FRACTION FORM,
IF WE COULD WRITE 32
WITH THE BASE OF 2
IT WOULD BE HELPFUL.
AND SINCE 32 IS = TO 2 x 16
AND 16 IS 2 TO THE FOURTH,
32 IS = TO 2 TO THE FIFTH,
ONE MORE FACTOR OF 2.
NOW, WE DO WANT TO HAVE
A COMMON BASE OF 2,
SO WE NEED TO CHANGE THE FORM
OF THIS FRACTION HERE.
REMEMBER WE CAN MOVE THIS UP
INTO THE NUMERATOR,
BUT IT WOULD CHANGE THE SIGN
OF THE EXPONENT.
SO 1/2 TO THE FOURTH IS = TO
2 RAISED TO THE POWER OF -4.
ALL THIS RAISED
TO THE POWER OF X
AND THIS IS = TO 2 TO THE FIFTH.
NOW, WE DO HAVE A COMMON BASE,
BUT JUST LIKE
AT THE PREVIOUS PROBLEM,
WE DO HAVE A POWER
RAISED TO A POWER HERE.
SO WE NEED TO MULTIPLY
THESE EXPONENTS,
SO WE'RE GOING TO HAVE 2 TO THE
POWER OF -4X = 2 TO THE FIFTH.
AND NOW THAT THESE ARE EQUAL
AND THE BASES ARE THE SAME
WE CAN CONCLUDE THAT -4X
MUST = 5
BECAUSE THE EXPONENTS
MUST ALSO BE EQUAL.
SO IF -4X = 5,
DIVIDE BOTH SIDES BY -4,
WHICH SIMPLIFIES TO X.
SO WE HAVE X = -5/4.
OKAY, WE'LL LOOK AT SOME MORE
EXAMPLES IN THE NEXT VIDEO.
