Simplify \ \textless \ br /\ \textgreater \ \sqrt[4]{16a^- 12}\ \textless \ br /\ \textgreater \ ​ ​​

Question

Simplify


`\ \textless \ br /\ \textgreater \ \sqrt[4]{16a^- 12}\ \textless \ br /\ \textgreater \ ​`

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Answer 1

`\huge\boxed{\sqrt[4]{16a^(-12)}=2a^(-3)=(2)/(a^3)}`

Explanation:

`16=2^4\\\\a^(-12)=a^((-3)(4))=\left(a^(-3)\right)^4\qquad\text{used}\ (a^n)^m=a^(nm)\\\\\sqrt[4]{16a^(-12)}=\bigg(16a^(-12)\bigg)^(1)/(4)\qquad\text{used}\ a^(1)/(n)=\sqrt[n]{a}\\\\=\bigg(2^4(a^(-3))^4\bigg)^(1)/(4)\qquad\text{use}\ (ab)^n=a^nb^n\\\\=\bigg(2^4\bigg)^(1)/(4)\bigg[(a^(-3))^4\bigg]^(1)/(4)\qquad\text{use}\ (a^n)^m=a^(nm)\\\\=2^{(4)((1)/(4))}(a^(-3))^{(4)((1)/(4))}=2^1(a^(-3))^1=2a^(-3)\qquad\text{use}\ a^(-n)=(1)/(a^n)`

`=2\left((1)/(a^3)\right)=(2)/(a^3)`