12 Fill in the blanks with \( < \), \( \rangle \), or \( = \) to correctly compare the expressions. \( \sqrt{3}+4 \quad 3+\sqrt{4} \) \( \sqrt{11} \sqrt[3]{90} \)

To compare the expressions \( \sqrt{3} + 4 \) and \( 3 + \sqrt{4} \): First, we simplify \( 3 + \sqrt{4} \) to \( 3 + 2 = 5 \). Now we compute \( \sqrt{3} + 4 \). \( \sqrt{3} \) is approximately 1.732, so \( \sqrt{3} + 4 \approx 1.732 + 4 = 5.732 \). Therefore, \( \sqrt{3} + 4 > 3 + \sqrt{4} \), and we can fill in the first blank with \( > \). Now, let's compare \( \sqrt{11} \) and \( \sqrt[3]{90} \): Calculating \( \sqrt[3]{90} \), we find that \( 90 \) is between \( 64 \) (which is \( 4^3 \)) and \( 125 \) (which is \( 5^3 \)), so \( \sqrt[3]{90} \) is between \( 4 \) and \( 5 \). Meanwhile, \( \sqrt{11} \) is approximately 3.317, which means \( \sqrt{11} < \sqrt[3]{90} \). We can fill in the second blank with \( < \). Putting it all together, we have: \( \sqrt{3}+4 \quad > \quad 3+\sqrt{4} \) \( \sqrt{11} \quad < \quad \sqrt[3]{90} \)