Question

Simplify-

(a) $\sqrt{32y^5}$

(b) $\sqrt[3]{54p^{10}}$

(c)$\sqrt[4]{64q^{10}}$

Short answer

The values are-

(a) $4y^2\sqrt{2y}$

(b) localid="1645704090408" $3p^3\sqrt[3]{2p}$

(c)$2q^2\sqrt{2q}$

Step-by-step solution

Step 1. Explanation

(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.

Step 2. Solving for part (a)

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

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

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

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

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

Step 3. Solving for part (b)

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 $\sqrt[3]{27p^9\times 2p}$.

Using he product rule, we get

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

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

Step 4. Solving for part (c)

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 $\sqrt[4]{16q^8\times 4q^2}$.

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

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