Write the following expression in simplest form. \( \sqrt{2^{5} \cdot 13^{3}} \) \( \sqrt{2^{5} \cdot 13^{3}}= \) (Simplify yourannswer. Type an exact answer, using radicals as needed.)

To simplify the expression \( \sqrt{2^{5} \cdot 13^{3}} \), follow these steps: 1. **Rewrite the Square Root as Exponents:** \[ \sqrt{2^{5} \cdot 13^{3}} = (2^{5} \cdot 13^{3})^{\frac{1}{2}} \] 2. **Apply the Exponent to Each Factor:** \[ = 2^{\frac{5}{2}} \cdot 13^{\frac{3}{2}} \] 3. **Simplify Each Term Separately:** - For \( 2^{\frac{5}{2}} \): \[ 2^{\frac{5}{2}} = 2^{2 + \frac{1}{2}} = 2^2 \cdot 2^{\frac{1}{2}} = 4\sqrt{2} \] - For \( 13^{\frac{3}{2}} \): \[ 13^{\frac{3}{2}} = 13^{1 + \frac{1}{2}} = 13 \cdot 13^{\frac{1}{2}} = 13\sqrt{13} \] 4. **Multiply the Simplified Terms:** \[ 4\sqrt{2} \cdot 13\sqrt{13} = (4 \times 13) \cdot (\sqrt{2} \times \sqrt{13}) = 52\sqrt{26} \] So, the expression in its simplest form is: \[ \sqrt{2^{5} \cdot 13^{3}} = 52\sqrt{26} \] **Answer:** \(52\,\sqrt{26}\)