1. **State the problem:** Find the square root of 4094 using the prime factorization method. 2. **Prime factorize 4094:** - Start dividing 4094 by the smallest prime numbers. - 4094 is even, so divide by 2: $$4094 \div 2 = 2047$$ - Check if 2047 is divisible by 2: no (it's odd). - Check divisibility by 3: sum of digits of 2047 is 2+0+4+7=13, not divisible by 3. - Check divisibility by 5: last digit not 0 or 5. - Check divisibility by 7: $$2047 \div 7 \approx 292.43$$ no. - Check divisibility by 11: alternating sum of digits is (2+4) - (0+7) = 6 - 7 = -1, not divisible by 11. - Check divisibility by 13: $$2047 \div 13 = 157.46$$ no. - Check divisibility by 17: $$2047 \div 17 = 120.41$$ no. - Check divisibility by 23: $$2047 \div 23 = 89$$ exactly. - So, 2047 = 23 \times 89. 3. **Prime factors of 4094:** $$4094 = 2 \times 23 \times 89$$ 4. **Check for pairs:** - For square root by prime factorization, factors must be in pairs. - Here, 2, 23, and 89 are all single primes with no pairs. 5. **Conclusion:** - Since no prime factors are repeated, 4094 is not a perfect square. - The square root cannot be simplified exactly by prime factorization. - Approximate square root: $$\sqrt{4094} \approx 63.99$$ **Final answer:** The square root of 4094 is not a perfect square and cannot be simplified by prime factorization. Its approximate value is $$63.99$$.