Let's simplify each square root expression step by step.
1. Simplifying \( \sqrt{121} \)
Step 1: Identify perfect squares. The number 121 is a perfect square because it can be expressed as \( 11 \times 11 \).
Step 2: Apply the square root:
\[ \sqrt{121} = \sqrt{11 \times 11} = 11 \]
Conclusion: The simplified form of \( \sqrt{121} \) is \( 11 \). Since 11 is a whole number, it is classified as a rational number.
2. Simplifying \( \sqrt{48} \)
Step 1: Factor 48 into its prime factors:
\[ 48 = 16 \times 3 = 4 \times 4 \times 3 = 2^4 \times 3^1 \]
Step 2: Identify perfect squares within the factorization. The perfect square is 16, which is \( 4 \times 4 \).
Step 3: Simplify using the square root property:
\[ \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3} \]
Conclusion: The simplified form of \( \sqrt{48} \) is \( 4\sqrt{3} \). Since \( \sqrt{3} \) is an irrational number, \( 4\sqrt{3} \) is also irrational. Thus, \( \sqrt{48} \) is irrational.
Final Answers:
- \( \sqrt{121} = 11 \) (rational)
- \( \sqrt{48} = 4\sqrt{3} \) (irrational)