What is the following product?
[tex]$\sqrt{12} \cdot \sqrt{18}$[/tex]
A. [tex]$\sqrt{30}$[/tex]
B. [tex]$5 \sqrt{6}$[/tex]
C. [tex]$6 \sqrt{5}$[/tex]
D. [tex]$6 \sqrt{6}$[/tex]
Answer (2)
The product 12 ⋅ 18 simplifies to 6 6 . This is done by multiplying the square roots, finding the product under a single square root, and simplifying. Therefore, the correct choice is option D: 6 6 .
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Multiply the square roots: 12 ⋅ 18 = 12 ⋅ 18 .
Calculate the product inside the square root: 12 ⋅ 18 = 216 , so we have 216 .
Simplify the square root: 216 = 36 ⋅ 6 = 6 6 .
The final answer is 6 6 .
Explanation
Understanding the Problem We are asked to find the product of two square roots, 12 and 18 , and simplify the result. Our goal is to express the product in its simplest form and match it with one of the given options.
Multiplying the Square Roots First, let's multiply the two square roots together. We know that a ⋅ b = a ⋅ b . Therefore, we have 12 ⋅ 18 = 12 ⋅ 18 Now, we need to calculate the product of 12 and 18.
Calculating the Product We can calculate 12 ⋅ 18 as follows: 12 ⋅ 18 = ( 2 ⋅ 6 ) ⋅ ( 3 ⋅ 6 ) = 6 ⋅ 6 ⋅ 2 ⋅ 3 = 36 ⋅ 6 = 216 So, we have 12 ⋅ 18 = 216 .
Simplifying the Square Root Now, we need to simplify 216 . We look for the largest perfect square that divides 216. We know that 216 = 36 ⋅ 6 , and 36 is a perfect square ( 36 = 6 2 ). Therefore, we can write 216 = 36 ⋅ 6 = 36 ⋅ 6 = 6 6 So, the simplified form of 216 is 6 6 .
Comparing with Options Finally, we compare our result, 6 6 , with the given options. We see that it matches the last option. Therefore, the product 12 ⋅ 18 simplifies to 6 6 .
Examples
Understanding how to simplify radicals is useful in many areas, such as physics and engineering, where you might need to calculate distances or areas. For example, if you are designing a square garden with an area of 216 m 2 , you would need to find the side length by calculating 216 , which simplifies to 6 6 meters. This skill also helps in simplifying expressions in algebra and calculus, making complex problems easier to solve.
