Question

Identify the equivalent expression for each of the expressions below. Root(5, (m + 2) ^ 3). Root(3, (m + 2) ^ 5). Root(5, m ^ 3) + 2. Root(3, m ^ 5) + 2

Answer 1

1.
`(\sqrt[5]{(m+2)})^(3) = (m+2)^{(3)/(5)}`

2.
`(\sqrt[3]{(m+2)})^(5) = (m+2)^{(5)/(3)}`

3.
`\sqrt[5]{(m)}^(3)+2 = m^{(3)/(5)}+2`

4.
`\sqrt[3]{(m)}^(5)+2 = m^{(5)/(3)}+2`

Explanation:

Recall that

`(\sqrt[n]{x})^(m) = (x^{(m)/(n)})`

Where
`x^(m)` is called radicand and n is called index

1. Root(5, (m + 2) ^ 3)

In this case,

n is 5

m is 3

x = (m + 2)

`(\sqrt[5]{(m+2)})^(3) = (m+2)^{(3)/(5)}`

2. Root(3, (m + 2) ^ 5)

In this case,

n is 3

m is 5

x = (m + 2)

`(\sqrt[3]{(m+2)})^(5) = (m+2)^{(5)/(3)}`

3. Root(5, m ^ 3) + 2

In this case,

n is 5

m is 3

x = m

`\sqrt[5]{(m)}^(3)+2 = m^{(3)/(5)}+2`

4. Root(3, m ^ 5) + 2

In this case,

n is 3

m is 5

x = m

`\sqrt[3]{(m)}^(5)+2 = m^{(5)/(3)}+2`