Bringing terms under a common radical sign (review)

Text

Bring under a single radical sign:

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

Solution

Each product is reduced to a single radical by bringing the factors to the common index (the least common multiple of the indices).

(a) Indices $3$ and $2$, l.c.m. $=6$: $\sqrt[3]{3^4}=\sqrt[6]{3^8}$ and $\sqrt{5^3}=\sqrt[6]{5^9}$, hence $\sqrt[3]{3^4}\cdot\sqrt{5^3}=\sqrt[6]{3^8\cdot 5^9}=3\cdot5\,\sqrt[6]{3^2\cdot5^3}=15\sqrt[6]{1125}.$

(b) $\sqrt[6]{5^3}\cdot\sqrt[6]{5^3}=\sqrt[6]{5^6}=5$ and $\sqrt[3]{7^3}=7$, so the product is an integer: $\sqrt[6]{5^3}\cdot\sqrt[6]{5^3}\cdot\sqrt[3]{7^3}=5\cdot7=35.$

(c) Indices $3,\,2,\,5$, l.c.m. $=30$: $\sqrt[3]{3^8}=\sqrt[30]{3^{80}}$, $\sqrt{2^7}=\sqrt[30]{2^{105}}$, $\sqrt[5]{7^9}=\sqrt[30]{7^{54}}$, hence $\sqrt[3]{3^8}\cdot\sqrt{2^7}\cdot\sqrt[5]{7^9}=\sqrt[30]{2^{105}\cdot3^{80}\cdot7^{54}}.$

$\boxed{(a)\ 15\sqrt[6]{1125}\qquad (b)\ 35\qquad (c)\ \sqrt[30]{2^{105}\cdot3^{80}\cdot7^{54}}}$