t Use the product rule to simplify the radical. sqrt (567) sqrt (567)=square (Type an exact answer in simplified form.)

To simplify the radical \(\sqrt{567}\) using the product rule, we first need to factorize 567 into its prime factors.<br /><br />1. **Factorize 567:**<br /> - 567 is divisible by 3 (since the sum of its digits, 5 + 6 + 7 = 18, is divisible by 3).<br /> - \(567 \div 3 = 189\)<br /> - 189 is also divisible by 3.<br /> - \(189 \div 3 = 63\)<br /> - 63 is also divisible by 3.<br /> - \(63 \div 3 = 21\)<br /> - 21 is also divisible by 3.<br /> - \(21 \div 3 = 7\)<br /><br />So, the prime factorization of 567 is:<br />\[ 567 = 3^4 \times 7 \]<br /><br />2. **Apply the product rule for radicals:**<br /> The product rule states that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\).<br /><br />Using this rule, we can write:<br />\[ \sqrt{567} = \sqrt{3^4 \times 7} \]<br /><br />3. **Simplify the expression:**<br /> We know that \(\sqrt{3^4} = 3^2 = 9\), because the square root of a power of 3 where the exponent is even simplifies directly.<br /><br />Thus,<br />\[ \sqrt{567} = \sqrt{3^4 \times 7} = \sqrt{3^4} \times \sqrt{7} = 9 \times \sqrt{7} \]<br /><br />Therefore, the simplified form of \(\sqrt{567}\) is:<br />\[ \boxed{9\sqrt{7}} \]