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Answer:
We conclude that:
[tex]\sqrt{\left(-81\right)x^2}=9ix[/tex]
Step-by-step explanation:
Given the radical expression
[tex]\sqrt{\left(-81\right)x^2}[/tex]
simplifying the expression
[tex]\sqrt{\left(-81\right)x^2}[/tex]
Remove parentheses: (-a) = -a
[tex]\sqrt{\left(-81\right)x^2}=\sqrt{-81x^2}[/tex]
Apply radical rule: [tex]\sqrt{-a}=\sqrt{-1}\sqrt{a},\:\quad \mathrm{\:assuming\:}a\ge 0[/tex]
[tex]=\sqrt{-1}\sqrt{81x^2}[/tex]
Apply imaginary number rule: [tex]\sqrt{-1}=i[/tex]
[tex]=i\sqrt{81x^2}[/tex]
Apply radical rule: [tex]\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b},\:\quad \mathrm{\:assuming\:}a\ge 0,\:b\ge 0[/tex]
[tex]=\sqrt{81}i\sqrt{x^2}[/tex]
[tex]=9i\sqrt{x^2}[/tex]
Apply radical rule: [tex]\sqrt[n]{a^n}=a,\:\quad \mathrm{\:assuming\:}a\ge 0[/tex]
[tex]=9ix[/tex]
Therefore, we conclude that:
[tex]\sqrt{\left(-81\right)x^2}=9ix[/tex]