Taking factors out of the radical sign (review II)

Text

Take out of the radical sign every possible power:

• (a) $\sqrt[15]{3^{50}\cdot3^{100}}$;
• (b) $\sqrt[13]{3^{17}\cdot5^{40}}$;
• (c) $\sqrt[5]{2^{13}\cdot5^{33}\cdot3^{22}}$.

Solution

Divide each exponent by the index: the quotient comes out, the remainder stays under the radical.

(a) $3^{50}\cdot3^{100}=3^{150}$ and $150$ is a multiple of $15$, so the root is exact: $\sqrt[15]{3^{150}}=3^{10}=59049.$

(b) Index $13$: $17=13+4$ and $40=39+1=3\cdot13+1$, so $3^{17}=3^{13}\cdot3^4$ and $5^{40}=(5^3)^{13}\cdot5$: $\sqrt[13]{3^{17}\cdot5^{40}}=3\cdot5^3\sqrt[13]{3^4\cdot5}=375\sqrt[13]{405}.$

(c) Index $5$: $13=10+3$, $33=30+3$, $22=20+2$, so $2^2$, $5^6$, $3^4$ come out: $\sqrt[5]{2^{13}\cdot5^{33}\cdot3^{22}}=2^2\cdot5^6\cdot3^4\sqrt[5]{2^3\cdot5^3\cdot3^2}=5062500\sqrt[5]{9000}.$

$\boxed{(a)\ 59049\qquad (b)\ 375\sqrt[13]{405}\qquad (c)\ 5062500\sqrt[5]{9000}}$