Question

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

Answer 1

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

Answer 2

Answer:

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

Step-by-step explanation:

Here, the given expression is,

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

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

$=(27a^3b^{6+1})^\frac{1}{3}$  

$=(3^3a^3b^6.b^1)^\frac{1}{3}$   ( $a^{m+n}=a^m.a^n$ )

$=((3ab^2)^3)^\frac{1}{3}.b^\frac{1}{3}$   ( $a^m.b^m=(ab)^m$ )

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