What is the value of $(\sqrt{5})^{-5/2} imes (\sqrt{5})^{-3/2}$ ?

Q: What is the value of $(\sqrt{5})^{-5/2} \times (\sqrt{5})^{-3/2}$ ?

To solve the expression $(\sqrt{5})^{-5/2} \times (\sqrt{5})^{-3/2}$, we need to apply the properties of exponents. 1. **Combine the Exponents:** The expression involves multiplying two terms with the same base, $\sqrt{5}$. When multiplying terms with the same base, we add the exponents. Therefore, we have: \[ (\sqrt{5})^{-5/2} \times (\sqrt{5})^{-3/2} = (\sqrt{5})^{(-5/2) + (-3/2)} \] 2. **Add the Exponents:** Calculate the sum of the exponents: \[ (-5/2) + (-3/2) = -8/2 = -4 \] 3. **Simplify the Expression:** Now, the expression becomes: \[ (\sqrt{5})^{-4} \] 4. **Convert to a Fraction:** Recall that $\sqrt{5} = 5^{1/2}$. Therefore: \[ (\sqrt{5})^{-4} = (5^{1/2})^{-4} = 5^{(-4 \times 1/2)} = 5^{-2} \] 5. **Evaluate the Negative Exponent:** A negative exponent indicates the reciprocal of the base raised to the positive exponent: \[ 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \] Thus, the value of the expression $(\sqrt{5})^{-5/2} \times (\sqrt{5})^{-3/2}$ is $\frac{1}{25}$, which corresponds to Option 1. **Explanation for Incorrect Options:** - **Option 2: $\frac{1}{125}$** - This would be the result if the expression evaluated to $5^{-3}$, which is not the case here. The correct exponent is $-2$, not $-3$. - **Option 3: $\frac{1}{325}$** - This option is incorrect because $325$ is not a power of $5$. The expression simplifies to a power of $5$, specifically $5^{-2}$. - **Option 4: $\frac{1}{625}$** - This would be the result if the expression evaluated to $5^{-4}$, which is not the case. The correct exponent is $-2$, not $-4$. Therefore, the correct answer is indeed Option 1: $\frac{1}{25}$.

Question

What is the value of $(\sqrt{5})^{-5/2} 	imes (\sqrt{5})^{-3/2}$ ?

Accepted Answer

To solve the expression $(\sqrt{5})^{-5/2} \times (\sqrt{5})^{-3/2}$, we need to apply the properties of exponents.

1. **Combine the Exponents:**
   The expression involves multiplying two terms with the same base, $\sqrt{5}$. When multiplying terms with the same base, we add the exponents. Therefore, we have:
   $$
   (\sqrt{5})^{-5/2} \times (\sqrt{5})^{-3/2} = (\sqrt{5})^{(-5/2) + (-3/2)}
   $$

2. **Add the Exponents:**
   Calculate the sum of the exponents:
   $$
   (-5/2) + (-3/2) = -8/2 = -4
   $$

3. **Simplify the Expression:**
   Now, the expression becomes:
   $$
   (\sqrt{5})^{-4}
   $$

4. **Convert to a Fraction:**
   Recall that $\sqrt{5} = 5^{1/2}$. Therefore:
   $$
   (\sqrt{5})^{-4} = (5^{1/2})^{-4} = 5^{(-4 \times 1/2)} = 5^{-2}
   $$

5. **Evaluate the Negative Exponent:**
   A negative exponent indicates the reciprocal of the base raised to the positive exponent:
   $$
   5^{-2} = \frac{1}{5^2} = \frac{1}{25}
   $$

Thus, the value of the expression $(\sqrt{5})^{-5/2} \times (\sqrt{5})^{-3/2}$ is $\frac{1}{25}$, which corresponds to Option 1.

**Explanation for Incorrect Options:**

- **Option 2: $\frac{1}{125}$**
  - This would be the result if the expression evaluated to $5^{-3}$, which is not the case here. The correct exponent is $-2$, not $-3$.

- **Option 3: $\frac{1}{325}$**
  - This option is incorrect because $325$ is not a power of $5$. The expression simplifies to a power of $5$, specifically $5^{-2}$.

- **Option 4: $\frac{1}{625}$**
  - This would be the result if the expression evaluated to $5^{-4}$, which is not the case. The correct exponent is $-2$, not $-4$.

Therefore, the correct answer is indeed Option 1: $\frac{1}{25}$.