Use the properties of rational exponents to simplify $$ \left(5^{1 / 2} \cdot 5^{-3 / 2}\right)^{-1 / 4} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression $$ \left(5^{1 / 2} \cdot 5^{-3 / 2}\right)^{-1 / 4} $$ using the properties of rational exponents, follow these steps:

1. **Combine the exponents inside the parentheses:**
   $$
   5^{1/2} \cdot 5^{-3/2} = 5^{1/2 + (-3/2)} = 5^{-1}
   $$

2. **Apply the outer exponent:**
   $$
   \left(5^{-1}\right)^{-1/4} = 5^{-1 \times (-1/4)} = 5^{1/4}
   $$

So, the simplified form of the expression is:
$$
5^{1/4}
$$
Alternatively, this can be written using radical notation as:
$$
\sqrt[4]{5}
$$

**Final Answer:** $$5^{\,1/4}$$
The simplified form of the expression is $$5^{1/4}$$ or $$\sqrt[4]{5}$$.

Use the properties of rational exponents to simplify