Q. 8.42 - Intermediate Algebra

Question

Simplify:

$\left(a\right) \sqrt{\frac{x^{14}}{x^{10}}} \left(b\right) \sqrt[3]{\frac{m^{13}}{m^7}} \left(c\right) \sqrt[5]{\frac{n^{12}}{n^2}}.$

Step-by-Step Solution

Answer

Part a. After simplifying we get, $x^2.$

Part b. After simplifying we get, $m^2.$

Part c. After simplifying we get, $n^2.$ 

Part a Step 1. Given Information

The given expression is $\sqrt{\frac{x^{14}}{x^{10}}}.$

We have to simplify it.

Part a Step 2. Use the Quotient Property of exponents to simplify

By using the property we get,

$$\begin{gathered} \sqrt{\frac{x^{14}}{x^{10}}} \\ =\sqrt{x^{14-10}} \\ =\sqrt{x^4} \\ =\sqrt{{\left(x^2\right)}^2} \\ =x^2 \end{gathered}$$

Part b Step 1. Given Information

The given expression is $\sqrt[3]{\frac{m^{13}}{m^7}}.$

We have to simplify it.

Part b Step 2. Use the Quotient Property of exponents to simplify

By using the property we get,

$$\begin{gathered} \sqrt[3]{\frac{m^{13}}{m^7}} \\ =\sqrt[3]{m^{13-7}} \\ =\sqrt[3]{m^6} \\ =\sqrt[3]{{\left(m^2\right)}^3} \\ =m^2 \end{gathered}$$

Part c Step 1. Given Information

The given expression is $\sqrt[5]{\frac{n^{12}}{n^2}}.$

We have to simplify it.

Part c Step 2. Use the Quotient Property of exponents to simplify

By using the property we get,

$$\begin{gathered} \sqrt[5]{\frac{n^{12}}{n^2}} \\ =\sqrt[5]{n^{12-2}} \\ =\sqrt[5]{n^{10}} \\ =\sqrt[5]{{\left(n^2\right)}^5} \\ =n^2 \end{gathered}$$