Use the product rule to simplify the radical. sqrt (756) sqrt (756)= square (Type an exact answer in simplified form.)

To simplify the radical \(\sqrt{756}\) using the product rule, we first need to factor 756 into its prime factors.<br /><br />1. **Factor 756:**<br /><br /> - 756 is even, so divide by 2:<br /> \[<br /> 756 \div 2 = 378<br /> \]<br /> - 378 is even, so divide by 2 again:<br /> \[<br /> 378 \div 2 = 189<br /> \]<br /> - 189 is divisible by 3 (since the sum of its digits, 18, is divisible by 3):<br /> \[<br /> 189 \div 3 = 63<br /> \]<br /> - 63 is also divisible by 3:<br /> \[<br /> 63 \div 3 = 21<br /> \]<br /> - 21 is divisible by 3:<br /> \[<br /> 21 \div 3 = 7<br /> \]<br /> - Finally, 7 is a prime number.<br /><br />So, the prime factorization of 756 is:<br />\[<br />756 = 2^2 \times 3^3 \times 7<br />\]<br /><br />2. **Apply the Product Rule for Radicals:**<br /><br />The product rule states that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). We can apply this rule to separate the perfect squares from the non-perfect squares:<br /><br />\[<br />\sqrt{756} = \sqrt{2^2 \times 3^3 \times 7}<br />\]<br /><br />Separate the perfect squares:<br />\[<br />= \sqrt{(2^2) \times (3^2) \times 3 \times 7}<br />\]<br /><br />Simplify the square roots of the perfect squares:<br />\[<br />= \sqrt{2^2} \times \sqrt{3^2} \times \sqrt{3 \times 7}<br />\]<br /><br />Calculate the square roots:<br />\[<br />= 2 \times 3 \times \sqrt{21}<br />\]<br /><br />Thus, the simplified form of \(\sqrt{756}\) is:<br />\[<br />6\sqrt{21}<br />\]<br /><br />Therefore, the exact answer in simplified form is:<br />\[<br />\boxed{6\sqrt{21}}<br />\]