Question

Help with Algebra 2: 1.
`\frac{\sqrt[7]{x^5} }{\sqrt[4]{x^2} }`

2.
`3\sqrt[4]{(x-2)^3} -4=20`

Answer 1

See explanation

Explanation:

1. Given the expression

`\frac{\sqrt[7]{x^5} }{\sqrt[4]{x^2} }`

Note that

`\sqrt[7]{x^5}=x^{(5)/(7)} \\ \\\sqrt[4]{x^2}=x^{(2)/(4)}=x^{(1)/(2)}`

When dividing
`\sqrt[7]{x^5}` by
`\sqrt[4]{x^2},` we have to subtract powers (we cannot subtract 4 from 7, because then we get another expression), so

`(5)/(7)-(2)/(4)=(5)/(7)-(1)/(2)=(5\cdot 2-1\cdot 7)/(14)=(3)/(14)`

and the result is
`x^{(3)/(14)}=\sqrt[14]{x^3}`

2. Given equation
`3\sqrt[4]{(x-2)^3} -4=20`

Add 4:

`3\sqrt[4]{(x-2)^3} -4+4=20+4\\ \\3\sqrt[4]{(x-2)^3}=24`

Divide by 3:

`\sqrt[4]{(x-2)^3} =8`

Rewrite the equation as:

`(x-2)^{(3)/(4)}=8\\ \\(x-2)^{(3)/(4)}=2^3`

Hence,

`\left((x-2)^{(3)/(4)}\right)^{(4)/(3)}=(2^3)^{(4)/(3)}\\ \\x-2=2^{3\cdot (4)/(3)}\\ \\x-2=2^4\\ \\x-2=16\\ \\x-2+2=16+2\\ \\x=18`