SOLUTION: Simplify each expression. Assume that all variables are positive when they appear. A. 9(24)^(1/3) - (81)^(1/3) B. (32x)^(1/4) + (2x^5)^(1/4)

Algebra ->  Expressions -> SOLUTION: Simplify each expression. Assume that all variables are positive when they appear. A. 9(24)^(1/3) - (81)^(1/3) B. (32x)^(1/4) + (2x^5)^(1/4)       Log On


   

Question 1208436: Simplify each expression. Assume that all variables are positive when they appear.
A. 9(24)^(1/3) - (81)^(1/3)
B. (32x)^(1/4) + (2x^5)^(1/4)


Found 3 solutions by Edwin McCravy, ikleyn, mccravyedwin:
Answer by Edwin McCravy(20059) About Me  (Show Source):
You can put this solution on YOUR website!
A. matrix%282%2C1%2C%22%22%2C9%2824%29%5E%281%2F3%29+-+%2881%29%5E%281%2F3%29%29

Break all the base numbers down to primes:

9=3%2A3=3%5E2
24=2%2A12=2%2A2%2A6=2%2A2%2A2%2A3=2%5E3%2A3
81=9%2A9=3%2A3%2A3%2A3=3%5E4



Multiply exponents to raise powers to powers:



matrix%282%2C1%2C%22%22%2C3%5E2%2A2%5E1%2A3%5E%281%2F3%29+-+3%5E%284%2F3%29%29%29

Add exponents of 3 to multiply powers of 3  2%2B1%2F3=7%2F3

matrix%282%2C1%2C%22%22%2C2%2A3%5E%287%2F3%29-3%5E%284%2F3%29%29%29

Factor out 34/3:
Divide powers of 3 by subtracting exponents:

matrix%282%2C1%2C%22%22%2C3%5E%284%2F3%29%282%2A3%5E%283%2F3%29-1%29%29

matrix%282%2C1%2C%22%22%2C3%5E%284%2F3%29%282%2A3%5E1-1%29%29

matrix%282%2C1%2C%22%22%2C3%5E%284%2F3%29%286-1%29%29

matrix%282%2C1%2C%22%22%2C3%5E%284%2F3%29%285%29%29

matrix%282%2C1%2C%22%22%2C5%2A3%5E%284%2F3%29%29

Try this one on your own:

B. (32x)^(1/4) + (2x^5)^(1/4)

Edwin

Answer by ikleyn(52799) About Me  (Show Source):
You can put this solution on YOUR website!
.
Simplify each expression. Assume that all variables are positive when they appear.
(A) 9(24)^(1/3) - (81)^(1/3)
(B) (32x)^(1/4) + (2x^5)^(1/4)
~~~~~~~~~~~~~~~~~~~ 

(A)  9(24)^(1/3) = 9*(2^3*3)^(1/3) = 9*2*(3^(1/3)) = 18*(3^(1/3)).


     (81)^(1/3) = (3^4)^(1/3)) = 3*(3^(1/3)).


     THEREFORE,  9(24)^(1/3) - (81)^(1/3) = 18*(3^(1/3)) - 3*(3^(1/3)) = 15*(3^(1/3)).    ANSWER


     CHECK using a calculator.  Left side  9(24)^(1/3) - (81)^(1/3) = (using my MS EXCEL) = 21.63374355...

                                Right side  15*(3^(1/3)) = (using my MS EXCEL) = 21.63374355...


     Both values are equal, HENCE, the answer is confirmed.

Completed to the end.

----------------------

In his solution,  although it is long,  Edwin did not simplify the given expression to the end.

Therefore,  I came to make this job complete in a way as it  SHOULD  be done.


////////////////////////


Regarding part  B,  you write,  " Assume that all variables are positive when they appear. "

It is not an assumption,  which one can make or do not make.

This condition  x >= 0  describes the  DOMAIN,  where the whole expression is defined.

It is not defined if   x < 0.


So,  it would be more accurate mathematically to write  " simplify expression in its domain,  where it is defined. "


!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!


Dear visitor,  if you're confused about the post by @mccravyedwin,  it is Edwin making a joke this way.

In other words,  Edwin is in a good mood and agrees with me.

Maybe even repentant.



Answer by mccravyedwin(407) About Me  (Show Source):
You can put this solution on YOUR website!

It depends on how you define the word "simplified".

matrix%282%2C1%2C%22%22%2C5%2A3%5E%284%2F3%29%29

requires only 4 digits to write.

Ikleyn's solution matrix%282%2C1%2C%22%22%2C15%2A3%5E%281%2F3%29%29

requires a whopping 5 digits.  J

Edwin