The Zero Exponent Propety is \( a^{0}=1, a \neq 0 \). 2. How could you use the properties of exponents to explan why \( a^{0}=1 \) ? b. How could you apply the propenies of exponents to evaluere \( 5^{\frac{0}{2}} \) ? a. Choose the correct answer below. A. \[ \begin{array}{l} a^{0}=a^{1-1} \\ a^{0}=a^{1} \cdot a^{1} \\ a^{1}=a^{1} \cdot a^{0} \\ \frac{a^{1}}{a^{1}}=a^{0} \\ 1=a^{0} \end{array} \] C. \( a^{0} \cdot a^{1}=a^{0+1} \) B. \[ \begin{aligned} a^{0} & =1 \\ a^{2}-a^{2}+1 & =1 \\ a^{2}-a^{2}+a^{0} & =a^{0} \\ 0+a^{0} & =a^{0} \\ a^{0} & =a^{0} \end{aligned} \] D. \( \quad a^{2} \cdot a^{3}=\left(a^{2}\right)^{3} \)

**Step 1. Explaining Why \( a^{0}=1 \) Using Exponent Properties** Recall the property of exponents: \[ \frac{a^{m}}{a^{n}}=a^{m-n} \] for any nonzero \( a \). Choose \( m = 1 \) and \( n = 1 \). Then: \[ \frac{a^{1}}{a^{1}}=a^{1-1}=a^{0} \] Since any nonzero number divided by itself equals 1, \[ \frac{a^{1}}{a^{1}}=1. \] Thus, we have: \[ a^{0}=1. \] Among the provided choices, option **A** follows this reasoning: \[ \begin{array}{l} a^{0}=a^{1-1} \\ \text{implying}\\ a^{0}=\dfrac{a^{1}}{a^{1}}=1. \end{array} \] So, option A is the correct choice. --- **Step 2. Evaluating \( 5^{\frac{0}{2}} \)** First note that \[ \frac{0}{2}=0. \] Thus, the expression becomes: \[ 5^{\frac{0}{2}}=5^{0}. \] Using the result from Step 1, we know that for any nonzero number, \[ 5^{0}=1. \] --- **Final Answers** - **a.** The correct reasoning is shown in option **A**. - **b.** \( 5^{\frac{0}{2}}=1 \).