Question

What is the simplest form of 3 sqrt 27a^3b^7?

Answer 1

`\sqrt[3]{27a^3b^7}\\\\\text{use}\ \sqrt[3]{xy}=\sqrt[3]{x}\cdot\sqrt[3]{y}\\\\=\sqrt[3]{27}\cdot\sqrt[3]{a^3}\cdot\sqrt[3]{b^7}=\sqrt[3]{3^3}\cdot√(a^3)\cdot\sqrt[3]{b^(3+3+1)}\\\\\text{use}\ \sqrt[3]{x^3}=x\ \text{and}\ a^n\cdot a^m=a^(n+m)\\\\=3\cdot a\cdot\sqrt[3]{b^3\cdot b^3\cdot b^1}\\\\\text{use}\ \sqrt[3]{xy}=\sqrt[3]{x}\cdot\sqrt[3]{y}\\\\=3a\cdot\sqrt[3]{b^3}\cdot\sqrt[3]{b^3}\cdot\sqrt[3]{b}\\\\\text{use}\ \sqrt[3]{x^3}=x\\\\=3a\cdot b\cdot b\cdot\sqrt[3]{b}=\boxed{3ab^2\sqrt[3]{b}}`

Answer 2

The simplest form is
`3ab^2\sqrt[3]{b}`

Explanation:

Here, the given expression is,

`\sqrt[3]{27a^3b^7}`

`=(27a^3b^7)^(1)/(3)` (
`\sqrt[n]{x} = x^(1)/(n)` )

`=(27a^3b^(6+1))^(1)/(3)`

`=(3^3a^3b^6.b^1)^(1)/(3)` (
`a^(m+n)=a^m.a^n` )

`=((3ab^2)^3)^(1)/(3).b^(1)/(3)` (
`a^m.b^m=(ab)^m` )

`=3ab^2.\sqrt[3]{b}` (
`(a^m)^n= a^(mn)` )