1. The problem is to multiply the mixed number $4 \frac{1}{4}$ by itself, i.e., calculate $4 \frac{1}{4} \times 4 \frac{1}{4}$. 2. First, convert the mixed number to an improper fraction. Recall that $a \frac{b}{c} = \frac{ac + b}{c}$. 3. For $4 \frac{1}{4}$, multiply the whole number 4 by the denominator 4 and add the numerator 1: $4 \times 4 + 1 = 16 + 1 = 17$. So, $4 \frac{1}{4} = \frac{17}{4}$. 4. Now multiply the fractions: $\frac{17}{4} \times \frac{17}{4} = \frac{17 \times 17}{4 \times 4} = \frac{289}{16}$. 5. Convert the improper fraction back to a mixed number by dividing 289 by 16. 6. $289 \div 16 = 18$ remainder $1$, so the mixed number is $18 \frac{1}{16}$. 7. Therefore, $4 \frac{1}{4} \times 4 \frac{1}{4} = 18 \frac{1}{16}$. Final answer: $18 \frac{1}{16}$.