=======Exercise 2.3 (Solutions)========
====Question 1====
Write each radical expression in exponential notation and each exponential expression in radical notation, Do not simplify.\\  
         *(i) $\sqrt[3]{-64}$	                            *(ii) $2^{35}$\\	           
         * (iii) $-7^\frac{1}{3}$                           * (iv) $y^\frac{-2}{3}$\\	

**Solution**\\

* (i) $\sqrt[3]{-64} = -64^\frac{1}{3}$				( Exponential form)
	
* (ii) $2^\frac{3}{5} = \sqrt[5]{2}^{3}$				        (Radical form)

* (iii) $-7^\frac{1}{3} = -\sqrt[3]{7}$				        (Radical form)

* (iv) $y^\frac{-2}{3} = \sqrt[3]{y}^{-2}$			        (Radical form)


====Question 2====
Tell whether the following statements are true or false\\
* (i)   $ 5^\frac{1}{5} = \sqrt{5}$\\		       * (ii)  $2^\frac{2}{3} = \sqrt[3]{4}$\\
* (iii) $\sqrt{49} = \sqrt{7}$\\                       * (iv)  $\sqrt[3]{x}^{27} = x^3$\\

**Solution**\\

  * (i)   False
  * (ii)  True
  * (iii) False
  * (iv)  False


====Question 3====
Simplify the following radical expression
  * (i) $\sqrt[3]{-125}$
  * (ii) $\sqrt[4]{32}$
  * (iii) $\sqrt[5]{\frac{3}{32}}$
  * (iv) $\sqrt[3]{\frac{-8}{27}}$

**Soluton**\\

 (i)    
$$\begin{array}{cl}
\sqrt[3]{-125} &= \sqrt[3]{-5^3}\\
& = {-5}^{3\times\frac{1}{3}}\\
&= {-5}
\end{array}$$

 (ii) 
$$\begin{array}{cl}
\sqrt[4]{32} &= \sqrt[4]{{2}^5}\\
&= \left(2^{4}\times{2}\right)^\frac{1}{4}\\
&= \left(2^{4})^\frac{1}{4}\right)\times\sqrt[4]{2}\\
&= 2\sqrt[4]{2}
\end{array}$$

 (iii) 
$$\begin{array}{cl}
\sqrt[5]{\frac{3}{32}} &= \left(\frac{3}{{2^5}}\right)^\frac{1}{5}\\
&= \frac{\sqrt[5]{3}}{2}
\end{array}$$                          

 (iv) 
$$\begin{array}{cl}
\sqrt[3]{\frac{-8}{27}} &= \sqrt[3]{\left(\frac{-2^3}{3^3}\right)}\\
&= \left(\frac{-2}{3}\right)^{3\times\frac{1}{3}}\\
&= \frac{-2}{3}
\end{array}$$