1. **Problem statement:** (a) Find the prime factor decomposition of 396 and express it in index form. (b) Using the prime factorization of 396, determine which two of the given numbers are factors of 396. 2. **Formula and rules:** Prime factorization means expressing a number as a product of prime numbers raised to their powers. If $a = p_1^{e_1} \times p_2^{e_2} \times \cdots \times p_n^{e_n}$, where $p_i$ are primes and $e_i$ are exponents, then any factor of $a$ must have prime factors with exponents less than or equal to those in $a$. 3. **Step-by-step solution for (a):** - Start dividing 396 by the smallest prime 2: $$396 \div 2 = 198$$ - Divide 198 by 2 again: $$198 \div 2 = 99$$ - 99 is not divisible by 2, try next prime 3: $$99 \div 3 = 33$$ - Divide 33 by 3 again: $$33 \div 3 = 11$$ - 11 is a prime number. - So the prime factors are: $2, 2, 3, 3, 11$ - Express in index form: $$396 = 2^2 \times 3^2 \times 11^1$$ 4. **Step-by-step solution for (b):** - Given numbers and their prime factorizations: - 88 = $2^3 \times 11$ - 14 = $2 \times 7$ - 9 = $3^2$ - 121 = $11^2$ - 22 = $2 \times 11$ - Check which are factors of 396 by comparing exponents: - For 88: $2^3$ vs $2^2$ in 396, exponent 3 is greater than 2, so 88 is **not** a factor. - For 14: contains prime 7 which is not in 396, so **not** a factor. - For 9: $3^2$ matches $3^2$ in 396, so 9 **is** a factor. - For 121: $11^2$ vs $11^1$ in 396, exponent 2 is greater, so **not** a factor. - For 22: $2^1 \times 11^1$ both exponents less or equal to those in 396, so 22 **is** a factor. **Final answers:** (a) $$396 = 2^2 \times 3^2 \times 11$$ (b) The two factors of 396 from the list are **9** and **22**.