Question

Which of the following numbers are greater than 6 and less than 8? Explainhow you know.

Answer 1

Let's analyze each case and see if they are less than 8 first:

`7<8,therefore(7)^{(1)/(2)}<8`
`\sqrt[]{64}=8`

thus

`\sqrt[]{60}<\sqrt[]{64}=8`
`\sqrt[]{60}<8`

Finally,

`64<80`
`\sqrt[]{64}<\sqrt[]{80}`

thus

`8<\sqrt[]{80}`

In sumary, the first two options are less than 8, but not the third.

Now let's see if they are greater than 6

`9>7`
`\sqrt[]{9}>\sqrt[]{7}`
`3>\sqrt[]{7}`

and

`6>3>\sqrt[]{7}`

thus

`6>\sqrt[]{7}`

Now

`36<60`
`\sqrt[]{36}<\sqrt[]{60}`

and so

`6<\sqrt[]{60}`

Finally

`\sqrt[]{80}>\sqrt[]{60}>6`

thus

`\sqrt[]{80}>6`

In conclussion, the second and third options are greater than 6, but not the first.