Question

Simply the expression in the simplest form​

Answer 1

`\frac{\sqrt[3]{20}}{5}`

Explanation:

Let
`x = \sqrt[3]{(4)/(25)}`, we proceed to show the procedure to determine the simplest form of this number:

1)
`\sqrt[3]{(4)/(25) }` Given.

2)
`\left((4)/(25)\right)^(1/3)` Definition of cubic root.

3)
`[4\cdot (25)^(-1)]^(1/3)` Definition of division.

4)
`(4)^(1/3)\cdot [(25)^(-1)]^(1/3)`
`a^(c)\cdot b^(c) = (a\cdot b)^(c)`

5)
`\{(4)^(1/3)\cdot [(25)^(-1)]^(1/3)\} \cdot \{[(25)^(-1)]^(2/3)\cdot [(25)^(-1)]^(-2/3)\}` Modulative and associative properties/Existence of multiplicative inverse/
`a^(b)\cdot a^(c) = a^(b+c)`

6)
`\{(4)^(1/3)\cdot [(25)^(-1)]^(-2/3)\}\cdot \{[(25)^(-1)]^(1/3)\cdot [(25)^(-1)]^(2/3)\}` Commutative and associative properties

7)
`\{(4)^(1/3)\cdot [(25)^(-1)]^(-2/3)\}\cdot (25)^(-1)`
`a^(b)\cdot a^(c) = a^(b+c)`

8)
`[(4)^(1/3)\cdot (25)^(2/3)]\cdot (25)^(-1)`
`(a^(b))^(c) = a^(b\cdot c)`

9)
`[4\cdot (25)^(2)]^(1/3)\cdot (25)^(-1)`
`(a^(b))^(c) = a^(b\cdot c)`/
`a^(c)\cdot b^(c) = (a\cdot b)^(c)`

10)
`(2500)^(1/3)\cdot (25)^(-1)` Definition of power and multiplication.

11)
`[(125)\cdot (20)]^(1/3)\cdot (25)^(-1)` Definition of multiplication.

12)
`(125)^(1/3)\cdot [(20)^(1/3)\cdot (25)^(-1)]`
`a^(c)\cdot b^(c) = (a\cdot b)^(c)`/Associative property.

13)
`5\cdot [(20)^(1/3)\cdot (25)^(-1)]` Definition of cubic root.

14)
`5\cdot [(20)^(1/3)\cdot (5)^(-1)\cdot (5)^(-1)]` Definition of multiplication/
`a^(c)\cdot b^(c) = (a\cdot b)^(c)`

15)
`[(20)^(1/3)\cdot (5)^(-1)][5\cdot (5)^(-1)]` Commutative and associative properties.

16)
`(20)^(1/3)\cdot (5)^(-1)` Existence of multiplicative inverse/Modulative property

17)
`\frac{\sqrt[3]{20}}{5}` Definitions of cubic root and division/Result