Question

`\sqrt{(-81)x^(2) }`

Answer 1

We conclude that:

`√(\left(-81\right)x^2)=9ix`

Explanation:

Given the radical expression

`√(\left(-81\right)x^2)`

simplifying the expression

`√(\left(-81\right)x^2)`

Remove parentheses: (-a) = -a

`√(\left(-81\right)x^2)=√(-81x^2)`

Apply radical rule:
`√(-a)=√(-1)√(a),\:\quad \mathrm{\:assuming\:}a\ge 0`

`=√(-1)√(81x^2)`

Apply imaginary number rule:
`√(-1)=i`

`=i√(81x^2)`

Apply radical rule:
`\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b},\:\quad \mathrm{\:assuming\:}a\ge 0,\:b\ge 0`

`=√(81)i√(x^2)`

`=9i√(x^2)`

Apply radical rule:
`\sqrt[n]{a^n}=a,\:\quad \mathrm{\:assuming\:}a\ge 0`

`=9ix`

Therefore, we conclude that:

`√(\left(-81\right)x^2)=9ix`