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- WE WANT TO SIMPLIFY THE CUBE ROOT OF -27
AND THE 5th ROOT OF -32.
BECAUSE THE NUMBER UNDERNEATH THE RADICAL OR THE RADICAND
IS NEGATIVE
THESE MAY SEEM MORE CHALLENGING
BUT AS LONG AS WE REMEMBER
THAT THE PRODUCT OF AN ODD NUMBER OF NEGATIVES
IS STILL NEGATIVE
WE'LL BE ABLE TO SIMPLIFY THESE.
SO FOR THE CUBE ROOT OF -27
IF WE IGNORE THE NEGATIVE SIGN FOR A MOMENT
AND RECOGNIZE THAT 27 IS A PERFECT CUBE
BECAUSE IT'S EQUAL TO 3 x 3 x 3
AND IF WE MADE THESE FACTORS NEGATIVE,
THE PRODUCT WOULD BE -27.
SO WE CAN REWRITE THE CUBE ROOT OF -27 AS -3 x -3 x -3,
AND BECAUSE WE'RE TAKING THE CUBE ROOT OF THIS
AND WE HAVE 3 EQUAL FACTORS
THIS WILL SIMPLIFY TO JUST ONE OF THE FACTORS, WHICH IS -3.
OF COURSE IF WE WANTED TO WE COULD REWRITE THIS
AS THE CUBE ROOT OF -3 RAISED TO THE 3rd
AND RECOGNIZE THAT THE CUBE ROOT AND THIS EXPONENT OF 3
WOULD UNDO EACH OTHER RESULTING IN -3.
BUT AGAIN, THE REASON WHY THE CUBE ROOT -27 = -3
IS BECAUSE -3 RAISED TO THE 3rd = -27.
SO FOR THE 5th ROOT OF -32,
AGAIN IGNORING THE NEGATIVE SIGN FOR A MOMENT,
IF WE WROTE THE PRIME FACTORIZATION OF 32,
IT'S EQUAL TO FIVE FACTORS OF 2.
AND IF WE MADE THESE 2s NEGATIVE THE PRODUCT WOULD BE -32.
SO AGAIN, WE COULD REWRITE THE 5th ROOT OF -32
AS FIVE FACTORS OF -2.
AND BECAUSE THE INDEX IS 5 AND WE HAVE FIVE EQUAL FACTORS
THIS SIMPLIFIES PERFECTLY TO ONE OF THE FACTORS OF -2.
OR AGAIN, IF WE WANTED TO WE COULD HAVE WRITTEN THIS
AS THE 5th ROOT OF -2 TO THE 5th
AND THE 5th ROOT AND THE EXPONENT OF 5 UNDO EACH OTHER
RESULTING IN -2.
AGAIN, THE REASON WHY THE 5th ROOT OF -32 = -2
IS BECAUSE -2 RAISED TO THE 5th IS EQUAL TO -32.
OKAY. HOPE YOU FOUND THIS HELPFUL.