Question

Could someone help me, please?

Answer 1

We can rewrite the expression under the radical as

`(81)/(16)a^8b^(12)c^(16)=\left(\frac32a^2b^3c^4\right)^4`

then taking the fourth root, we get

`\sqrt[4]{\left(\frac32a^2b^3c^4\right)^4}=\left|\frac32a^2b^3c^4\right|`

Why the absolute value? It's for the same reason that

`√(x^2)=|x|`

since both
`(-x)^2` and
`x^2` return the same number
`x^2`, and
`|x|` captures both possibilities. From here, we have

`\left|\frac32a^2b^3c^4\right|=\left|\frac32\right|\left|a^2\right|\left|b^3\right|\left|c^4\right|`

The absolute values disappear on all but the
`b` term because all of
`\frac32`,
`a^2` and
`c^4` are positive, while
`b^3` could potentially be negative. So we end up with

`\frac32a^2\left|b^3\right|c^4=\frac32a^2|b|^3c^4`