Bringing factors under a common radical

Text

Bring everything under a single radical sign:

• (a) $3^3\cdot\sqrt{2^5}$
• (b) $2^3\cdot\sqrt[3]{3^2}$
• (c) $5^3\cdot\sqrt[4]{2^3}\cdot\sqrt{7^8}$
• (d) $\sqrt[4]{4^9}\cdot\sqrt[5]{2^6}\cdot\sqrt[3]{3^2}$

Solution

To bring a factor $a$ under a radical of index $n$, rewrite it as $\sqrt[n]{a^{n}}$; to merge radicals of different indices, reduce to the common index (l.c.m.).

(a) $3^3=\sqrt{3^6}$: $\sqrt{3^6\cdot 2^5}=\sqrt{729\cdot 32}=\sqrt{23328}$.

(b) $2^3=\sqrt[3]{2^9}$: $\sqrt[3]{2^9\cdot 3^2}=\sqrt[3]{512\cdot 9}=\sqrt[3]{4608}$.

(c) $\sqrt{7^8}=7^4$, so $5^3\cdot 7^4=\sqrt[4]{5^{12}\cdot 7^{16}}$: $\sqrt[4]{5^{12}\cdot 7^{16}\cdot 2^{3}}$.

(d) $4^9=2^{18}$, hence $\sqrt[4]{2^{18}}=2^{9/2}$; common index $60$: $\sqrt[4]{2^{18}}=\sqrt[60]{2^{270}}$, $\sqrt[5]{2^{6}}=\sqrt[60]{2^{72}}$, $\sqrt[3]{3^{2}}=\sqrt[60]{3^{40}}$, giving $\sqrt[60]{2^{342}\cdot 3^{40}}$. Reducing the index by $2$: $\sqrt[30]{2^{171}\cdot 3^{20}}=2^{5}\sqrt[30]{2^{21}\cdot 3^{20}}$.

$\boxed{\;(a)\;\sqrt{23328}\qquad (b)\;\sqrt[3]{4608}\qquad (c)\;\sqrt[4]{5^{12}\cdot 7^{16}\cdot 2^{3}}\qquad (d)\;\sqrt[60]{2^{342}\cdot 3^{40}}=2^{5}\sqrt[30]{2^{21}\cdot 3^{20}}\;}$