Q. 8.41 - Intermediate Algebra

Question

Simplify:

$\left(a\right) \sqrt{\frac{a^8}{a^6}} \left(b\right) \sqrt[4]{\frac{x^7}{x^3}} \left(c\right) \sqrt[4]{\frac{y^{17}}{y^5}}$

Step-by-Step Solution

Answer

Part a. After simplifying we get, $a.$

Part b. After simplifying we get, $x.$

Part c. After simplifying we get, $y^3.$

Part a Step 1. Given Information

The given expression is $\sqrt{\frac{a^8}{a^6}}.$

We have to simplify it.

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

By using the property we get,

$$\begin{gathered} \sqrt{\frac{a^8}{a^6}} \\ =\sqrt{a^{8-6}} \\ =\sqrt{a^2} \\ =a \end{gathered}$$

Part b Step 1. Given Information

The given expression is $\sqrt[4]{\frac{x^7}{x^3}}.$

We have to simplify it.

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

By using the property we get,

$$\begin{gathered} \sqrt[4]{\frac{x^7}{x^3}} \\ =\sqrt[4]{x^{7-3}} \\ =\sqrt[4]{x^4} \\ =x \end{gathered}$$

Part c Step 1. Given Information

The given expression is $\sqrt[4]{\frac{y^{17}}{y^5}}.$

We have to simplify it.

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

By using the property we get,

$$\begin{gathered} \sqrt[4]{\frac{y^{17}}{y^5}} \\ =\sqrt[4]{y^{17-5}} \\ =\sqrt[4]{y^{12}} \\ =\sqrt[4]{{\left(y^3\right)}^4} \\ =y^3 \end{gathered}$$