91Ó°ÊÓ

(a) We will write the radical in form of product of two radicals with one being the largest perfect square factor of the number. Then we will simplify it to get the value.

(b) We will write the radical in form of product of two radicals with one being the largest perfect cube factor of the number. Then we will simplify it to get the value.

(c) We will write the radical in form of product of two radicals with one being the largest perfect fourth power factor of the number. Then we will simplify it to get the value.

We have $\sqrt{32y^5}$.

We have $16y^4$as the greatest perfect square factor of $32y^5$.

So this can be written as $\sqrt{16y^4×2y}$.

Using the product rule, we get role="math" localid="1644761116098" $$\begin{gathered} =\sqrt{16y^4}×\sqrt{2y} \\ =\sqrt{{\left(4y^2\right)}^2}×\sqrt{2y} \\ =4\left|y^2\right|\sqrt{2y} \end{gathered}$$

Thus we get the value asrole="math" localid="1644761128268" $4\left|y^2\right|\sqrt{2y}$.

We have $\sqrt[3]{54p^{10}}$.

We have role="math" localid="1644760785088" $27p^9$as the perfect cube factor of $54p^{10}$.

So this can be written as $\sqrt[3]{27p^9×2p}$.

Using he product rule, we get

$$\begin{gathered} =\sqrt[3]{27p^9}×\sqrt[3]{2p} \\ =\sqrt[3]{{\left(3p^3\right)}^2}×\sqrt[3]{2p} \\ =3p^3\sqrt[3]{2p} \end{gathered}$$

Thus we get the value as$3p^3\sqrt[3]{2p}$.

We have $\sqrt[4]{64q^{10}}$.

We have $16q^8$as the greatest perfect fourth power factor of $64q^{10}$.

So this can be written as $\sqrt[4]{16q^8×4q^2}$.

Using the product rule, we get
localid="1645704211099" $$\begin{gathered} =\sqrt[4]{16q^8}×\sqrt[4]{4q^2} \\ =\sqrt[4]{{\left(2q^2\right)}^4}×{\left(2q\right)}^{2×\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.} \\ =2q^2×{\left(2q\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} \\ =2q^2\sqrt{2q} \end{gathered}$$

Thus we get the value as localid="1645704220527" $2q^2\sqrt{2q}$.