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)

d. millimeters per hour.

step-by-step explanation:

we are given that,

carlo used a rain gauge to measure the amount of water his lawn is receiving from the sprinklers.

since, according to the options, we see that,

water cannot be measured by 'miles per hour', 'meters per second' and 'grams per seconds'.

we get that, the water can be measured by 'millimeters per hour'.

hence, the appropriate rate for describing the amount of water is 'millimeters per hour'.