Simplifying expressions with radicals and powers

Text

Simplify as much as possible using the properties of powers:

• (a) $\sqrt{3^3\cdot\sqrt[5]{3}}$
• (b) $\sqrt{2^7}\,\sqrt[4]{2^3}$
• (c) $\dfrac{\sqrt[3]{5^2}}{\sqrt[5]{5^4}}$
• (d) $\sqrt{2\cdot\sqrt{2\cdot\sqrt{2}}}$
• (e) $\sqrt{12}\,\sqrt[3]{36}$
• (f) $\sqrt{24\cdot\sqrt{6}}$

Solution

Rewrite everything with fractional exponents and use $\sqrt[n]{a}=a^{1/n}$, $a^m\cdot a^n=a^{m+n}$, $(a^m)^n=a^{mn}$.

(a) Inside: $3^3\cdot 3^{1/5}=3^{16/5}$; square root: $3^{8/5}=3\cdot 3^{3/5}=3\sqrt[5]{27}$.

(b) $2^{7/2}\cdot 2^{3/4}=2^{14/4+3/4}=2^{17/4}=2^4\cdot 2^{1/4}=16\sqrt[4]{2}$.

(c) $5^{2/3-4/5}=5^{(10-12)/15}=5^{-2/15}=\dfrac{1}{\sqrt[15]{25}}$.

(d) Nested radicals: $\sqrt{2}=2^{1/2}$, $2\cdot 2^{1/2}=2^{3/2}$, $\sqrt{2^{3/2}}=2^{3/4}$, $2\cdot 2^{3/4}=2^{7/4}$, $\sqrt{2^{7/4}}=2^{7/8}=\sqrt[8]{128}$.

(e) $12=2^2\cdot 3$, $36=2^2\cdot 3^2$; common index $6$: $\sqrt[6]{12^3\cdot 36^2}=\sqrt[6]{2^{10}\cdot 3^{7}}=2\cdot 3\,\sqrt[6]{2^{4}\cdot 3}=6\sqrt[6]{48}$.

(f) $24=2^3\cdot 3$; $\sqrt{24\cdot 6^{1/2}}=24^{1/2}\cdot 6^{1/4}$; common index $4$: $\sqrt[4]{24^2\cdot 6}=\sqrt[4]{2^{7}\cdot 3^{3}}=2\sqrt[4]{2^{3}\cdot 3^{3}}=2\sqrt[4]{216}$.

$\boxed{\;(a)\;3\sqrt[5]{27}\qquad (b)\;16\sqrt[4]{2}\qquad (c)\;\dfrac{1}{\sqrt[15]{25}}\qquad (d)\;\sqrt[8]{128}\qquad (e)\;6\sqrt[6]{48}\qquad (f)\;2\sqrt[4]{216}\;}$