Question

Simplify the expression.

a)$\sqrt[3]{{\left(x^{3}y\right)}^2{y}^4}$

b)$\sqrt{x^2{y}^4}$

Short answer

a) Answer is$\sqrt[3]{{\left(x^{3}y\right)}^2{y}^4}=x^2{y}^2$

b) Answer is$\sqrt{x^2{y}^4}=x{y}^2$

Step-by-step solution

Part a. Step 1. Identities of powers.

• ${\left(x^a\right)}^b=x^{ab}$
  
• $x^a\times x^b=x^{a+b}$
• $\sqrt[3]{x}=x^{\frac{1}{3}}$

Part a. Step 2. Re write the expression after applying the identities.

$$\begin{gathered} \sqrt[3]{{\left(x^{3}y\right)}^2{y}^4}=\sqrt[3]{\left(x^6\right)\left(y^2{y}^4\right)} \\ =\sqrt[3]{\left(x^6\right)\left(y^{2+4}\right)} \\ =\sqrt[3]{\left(x^6\right)\left(y^6\right)} \end{gathered}$$

Part a. Step 3. Simplify the expression by solving the powers.

$$\begin{gathered} \sqrt[3]{\left(x^6\right)\left(y^6\right)}=x^{6/3}{y}^{6/3} \\ =x^2{y}^2 \\ ={\left(xy\right)}^2 \end{gathered}$$

Part b. Step 1. Identities of powers.

• ${\left(x^a\right)}^b=x^{ab}$
  
• $x^a\times x^b=x^{a+b}$
• $\sqrt{x}=x^{\frac{1}{2}}$

Part b. Step 2. Re write the expression after applying the identities.

$\sqrt{x^2{y}^4}=x^{2/2}{y}^{4/2}$

Part b. Step 3. Simplify the expression by solving the powers.

$$\begin{gathered} x^{2/2}{y}^{4/2}=x^1{y}^2 \\ =x{y}^2 \end{gathered}$$