Answer:
[tex]\frac{\sqrt[3]{20}}{5}[/tex]
Step-by-step explanation:
Let [tex]x = \sqrt[3]{\frac{4}{25}}[/tex], we proceed to show the procedure to determine the simplest form of this number:
1) [tex]\sqrt[3]{\frac{4}{25} }[/tex] Given.
2) [tex]\left(\frac{4}{25}\right)^{1/3}[/tex] Definition of cubic root.
3) [tex][4\cdot (25)^{-1}]^{1/3}[/tex] Definition of division.
4) [tex](4)^{1/3}\cdot [(25)^{-1}]^{1/3}[/tex] [tex]a^{c}\cdot b^{c} = (a\cdot b)^{c}[/tex]
5) [tex]\{(4)^{1/3}\cdot [(25)^{-1}]^{1/3}\} \cdot \{[(25)^{-1}]^{2/3}\cdot [(25)^{-1}]^{-2/3}\}[/tex] Modulative and associative properties/Existence of multiplicative inverse/[tex]a^{b}\cdot a^{c} = a^{b+c}[/tex]
6) [tex]\{(4)^{1/3}\cdot [(25)^{-1}]^{-2/3}\}\cdot \{[(25)^{-1}]^{1/3}\cdot [(25)^{-1}]^{2/3}\}[/tex] Commutative and associative properties
7) [tex]\{(4)^{1/3}\cdot [(25)^{-1}]^{-2/3}\}\cdot (25)^{-1}[/tex] [tex]a^{b}\cdot a^{c} = a^{b+c}[/tex]
8) [tex][(4)^{1/3}\cdot (25)^{2/3}]\cdot (25)^{-1}[/tex] [tex](a^{b})^{c} = a^{b\cdot c}[/tex]
9) [tex][4\cdot (25)^{2}]^{1/3}\cdot (25)^{-1}[/tex] [tex](a^{b})^{c} = a^{b\cdot c}[/tex]/[tex]a^{c}\cdot b^{c} = (a\cdot b)^{c}[/tex]
10) [tex](2500)^{1/3}\cdot (25)^{-1}[/tex] Definition of power and multiplication.
11) [tex][(125)\cdot (20)]^{1/3}\cdot (25)^{-1}[/tex] Definition of multiplication.
12) [tex](125)^{1/3}\cdot [(20)^{1/3}\cdot (25)^{-1}][/tex] [tex]a^{c}\cdot b^{c} = (a\cdot b)^{c}[/tex]/Associative property.
13) [tex]5\cdot [(20)^{1/3}\cdot (25)^{-1}][/tex] Definition of cubic root.
14) [tex]5\cdot [(20)^{1/3}\cdot (5)^{-1}\cdot (5)^{-1}][/tex] Definition of multiplication/[tex]a^{c}\cdot b^{c} = (a\cdot b)^{c}[/tex]
15) [tex][(20)^{1/3}\cdot (5)^{-1}][5\cdot (5)^{-1}][/tex] Commutative and associative properties.
16) [tex](20)^{1/3}\cdot (5)^{-1}[/tex] Existence of multiplicative inverse/Modulative property
17) [tex]\frac{\sqrt[3]{20}}{5}[/tex] Definitions of cubic root and division/Result
