SOLUTION: Please Help, I am confused about how to work with third roots. I am supposed to multiply then simplify by factoring. (3rd root)of y^7 * (3rd root) of 16y^8 Use Rational exp

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Please Help, I am confused about how to work with third roots. I am supposed to multiply then simplify by factoring. (3rd root)of y^7 * (3rd root) of 16y^8 Use Rational exp      Log On


   

Question 303313: Please Help, I am confused about how to work with third roots.
I am supposed to multiply then simplify by factoring.
(3rd root)of y^7 * (3rd root) of 16y^8
Use Rational exponents to simplify the third root of x^6
Find the 16th root of (-8)^16
Rewrite the 4th root of 23 with a rational exponent.
Simplify the 3rd root of -1/64
Simplify by taking roots of the numerator and denominator
(3rd root) of 1000x^14/y^3
Solve
The square root of 5x+29 = x+3
I am completely lost when working with roots. Any help would be greatly appreciated!
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
multiply then simplify by factoring.
Use this as a cube root 3sqrt%28n%29 symbol
:
(3rd root)of y^7 * (3rd root) of 16y^8
3sqrt%28y%5E7%29+%2A+3sqrt%2816y%5E8%29
:
We can combine these under single radical
3sqrt%2816y%5E8y%5E7%29
:
Add exponents when you multiply
3sqrt%2816y%5E15%29
:
Factor to reveal perfect cubes
3sqrt%288%2A2%2Ay%5E3%2Ay%5E3%2Ay%5E3%2Ay%5E3%2Ay%5E3%29
:
Extract those perfect cubes
2%2Ay%5E5%2A3sqrt%282%29
:
:
Use Rational exponents to simplify the third root of x^6
to raise on power to another power, multiply exponents
%28x%5E6%29%5E%281%2F3%29 = x%5E%286%2A%281%2F3%29%29 = x%5E2
:
:
Find the 16th root of (-8)^16
%28%28-8%29%5E16%29%5E%281%2F16%29 = %28-8%29%5E%2816%2A%281%2F16%29%29 = %28-8%29%5E1 = -8
:
:
Rewrite the 4th root of 23 with a rational exponent.
23%5E%281%2F4%29
:
:
Simplify the 3rd root of -1/64
3sqrt%28-1%2F64%29 = 3sqrt%28%28-1%2F4%29%2A%28-1%2F4%29%2A%28-1%2F4%29%29 = %28-1%2F4%29
:
:
Simplify by taking roots of the numerator and denominator
(3rd root) of 1000x^14/y^3
3sqrt%281000x%5E14%2Fy%5E3%29
Extract cube root 1/y^3
%281%2Fy%29%2A3sqrt%281000x%5E14%29
Extract the cube root of 1000
%2810%2Fy%29%2A3sqrt%28x%5E14%29
Factor inside the radical
%2810%2Fy%29%2A3sqrt%28x%5E3%2Ax%5E3%2Ax%5E3%2Ax%5E3%2Ax%5E2%29
Extract
%2810%2Fy%29%2Ax%5E4%2A3sqrt%28x%5E2%29
:
;
Solve
The square root of 5x+29 = x+3
sqrt%285x%2B29%29+=+x%2B3
square both sides:
5x + 29 = (x+3)^2
FOIL the right side
5x + 29 = x^2 + 6x + 9
combine like terms on the right
0 = x^2 + 6x - 5x + 9 - 29
A quadratic equation
x^2 + x - 20 = 0
Factors to
(x+5)(x-4) = 0
two solutions
x = -5
and
x = +4
:
Check both solution in original equation
x = -5
sqrt%285%29-5%29%2B29%29+=+-5%2B3
sqrt%28-25%2B29%29+=+-2
sqrt%284%29+=+-2 not a solution
x = 4
sqrt%285%284%29%2B29%29+=+4%2B3
sqrt%2820%2B29%29+=+7
sqrt%2849%29+=+7; a good solution (x=4)