Math hw, simplify radical expressions, show all work.

12.  7xy ∛(xy^2)

13.  3x^2|y| sqrt(7x)  

14. 5  x^4   y ^4 y ∛(y^2)

Step-by-step explanation:

12.  (343 x^4y^5) ^ 1/3

separate  

343^1/3  x^4 ^ 1/3    y^5 ^ 1/3

using the power of power we can multiply the powers

343 ^ 1/3  x^ 4/3  y ^ 5/3

simplify

7   x^ 4/3  y ^ 5/3

if the power is greater than 1 we can split it

7 x x^ 1/3  y y ^ 2/3

7xy ( xy^2 ) ^ 1/3

7xy ∛(xy^2)

13.  (189 x^5y^6)^ 1/2/(3y^4)^ 1/2

combine

( (189 x^5y^6)/(3y^4))^ 1/2

(189/3  * x^5 * y^6/y^4) ^ 1/2

when we divide exponents with the same base, we subtract the exponent

(63 * x^5 * y^2) ^ 1/2

split

63 ^ 1/2   x^ 5 ^ 1/2  y ^ 2 ^ 1/2

sqrt (63) * sqrt * (x^5) * sqrt(y^2)

sqrt(9*7) sqrt( x^4 *x)  * sqrt(y^2)

sqrt(9) * sqrt(7)  sqrt(x^4) sqrt(x)  sqrt(y^2)

we need to make sure to take the positive value of sqrt(y^2)

3sqrt(7)  x^2  sqrt(x)  | y|  

3x^2|y| sqrt(7x)  

14.  (625 x^17 y^16) ^1/3  / (5 x^5 y^2) ^ 1/3

combine

(625 x^17 y^16  / 5 x^5 y^2) ^ 1/3

(625/5 * x^17/x^5   * y^16/y^2) ^ 1/3

when we divide exponents with the same base, we subtract the exponent

(125 x^(17-5)  y^(16-2)) ^ 1/3

(125 x^12 y ^ 14) ^ 1/3

split

125 ^ 1/3   x^ 12 ^ 1/3    y^ 14 ^ 1/3

power to the power ( multiply the power)

125 ^ 1/3 x^ 4 y ^ 14/3

5  x^4 y ^ 14/3

if the power is greater than 1 we can split it

5  x^4   y ^4 y ^2/3

5  x^4   y ^4 y ∛(y^2)

the fist one is not equalvalent but i think th second one isand the third isnt definetly or the last one is i think. hope this !

step-by-step explanation: