Simplify.
$\sqrt{12} \cdot \sqrt{6}$
Answer (1)
Multiply the numbers under the square roots: 12 × 6 = 12 × 6 = 72 .
Find the prime factorization of 72: 72 = 2 3 × 3 2 = 2 × 36 .
Simplify the square root by extracting perfect squares: 72 = 36 × 2 = 6 2 .
The simplified expression is 6 2 .
Explanation
Understanding the problem We are asked to simplify the expression 12 ⋅ 6 . Both terms are square roots of positive integers. We can use the property a ⋅ b = a ⋅ b .
Using the property of square roots Using the property a ⋅ b = a ⋅ b , we can rewrite the expression as 12 ⋅ 6 .
Calculating the product Now, we calculate the product 12 ⋅ 6 = 72 . So, the expression becomes 72 .
Simplifying the square root To simplify 72 , we find the prime factorization of 72. 72 = 2 3 ⋅ 3 2 = 2 ⋅ 2 2 ⋅ 3 2 = 2 ⋅ 4 ⋅ 9 . Therefore, 72 = 2 ⋅ 36 = 2 ⋅ 6 2 = 6 2 .
Final Answer Thus, the simplified expression is 6 2 .
Examples
Square roots appear in many contexts, such as calculating distances using the Pythagorean theorem. For example, if you have a right triangle with legs of length 12 and 6 , the length of the hypotenuse would be ( 12 ) 2 + ( 6 ) 2 = 12 + 6 = 18 = 3 2 . Simplifying radical expressions helps in finding exact values in geometry and other fields.
