Q2. - Algebra 2

Question

Determine whether$x^y\cdot x^z=x^{yz}$  , is sometimes, always or never true. Explain.

Step-by-Step Solution

Answer

The given expression is sometimes true for which the exponents terms $y=z=2$ and always true for which the exponents terms $yz=y+z$forall $\left\{y,z\right\}\in Z^{+}$.

Step 1. Write down the given information.

The given expression is$x^y\cdot x^z=x^{yz}$  .

Step 2. Explanation.

The given expression 

    $x^y\cdot x^z=x^{yz}$                                                               

 is simplified as:

 $\begin{array}{l}{x}^y\cdot x^z=x^{yz}\mathrm{....}\left(\text{Given}\right)\\ \text{But, }{x}^y\cdot x^z=x^{y+z}\mathrm{....}(\because x^m\cdot x^n=x^{m+n})\end{array}$                                             

Therefore, the given expression 

   ${\mathit{x}}^{\mathbf{y}}\mathbf{\cdot }{\mathit{x}}^{\mathbf{z}}\mathbf{=}{\mathit{x}}^{\mathbf{y}\mathbf{z}}$                                                                

 is not always true.

It  $x^y\cdot x^z=x^{yz}$

 will always be true when 

       $yz=y+z$                                                             

.

The expression 

   $x^y\cdot x^z=x^{yz}$                                                                

 is sometimes true for 

     $y=z=2$                                                                

Step 3. Conclusion.

The given expression is sometimes true for which the exponents terms $y=z=2$  and always true for which the exponents terms $yz=y+z\forall \{y,z\}\in Z^{+}$ .