- WE WANT TO SOLVE THE GIVEN
EXPONENTIAL EQUATION
USING THE PROPERTIES
OF EXPONENTS.
THE FIRST THING
WE SHOULD RECOGNIZE HERE IS THAT
WE DO HAVE A COMMON BASE OF 5.
SO, LET'S START BY SIMPLIFYING
THE LEFT SIDE OF THE EQUATION.
LET'S START BY SIMPLIFYING
THIS PRODUCT HERE.
REMEMBER THE RULE IS,
WHEN WE'RE MULTIPLYING
AND THE BASES ARE THE SAME,
WE ADD OUR EXPONENTS.
SO, THIS WOULD BE 5 TO THE POWER
OF X PLUS NEGATIVE 6
OR JUST X MINUS 6,
DIVIDED BY 5 TO THE 8TH
EQUALS 5 TO THE 12TH
AND NOW WE CAN SIMPLIFY
THIS QUOTIENT AGAIN
BECAUSE WE HAVE A COMMON BASE.
WHEN DIVIDING,
WE SUBTRACT THE EXPONENTS
AND IT'S ALWAYS THE EXPONENT
OF THE NUMERATOR
MINUS THE EXPONENT
OF THE DENOMINATOR.
SO, WE'D HAVE 5
TO THE POWER OF X MINUS 6
AND THEN, MINUS 8
EQUALS 5 TO THE 12TH.
THIS BECOMES 5 TO THE POWER OF X
MINUS 14 EQUALS 5 TO THE 12TH.
AND NOW IF THE LEFT SIDE
IS EQUAL TO THE RIGHT SIDE
AND WE HAVE A COMMON BASE OF 5,
THE EXPONENTS MUST BE EQUAL
TO EACH OTHER,
MEANING, X MINUS 14
MUST EQUAL 12.
SO, IF X MINUS 14 MUST EQUAL 12,
THEN WE CAN ADD 14
TO BOTH SIDES OF THE EQUATION
AND WE HAVE OUR SOLUTION X
EQUALS 26.
