Final answer:
The multiplication of the radicals 3√2· 2√2· 5√8· √18 simplifies to 60√3 after multiplying coefficients and radicands, extracting perfect squares, and reducing common factors. The correct answer is 60√3, which is choice D.
Step-by-step explanation:
The student asked to multiply the following radicals: 3√2· 2√2· 5√8· √18, and then simplify the result into its simplest radical form. To begin with the multiplication, we combine all the coefficients (numbers outside the radical) and all the radicands (numbers inside the radical) separately, as per the associativity of multiplication. Here the coefficients are 3, 2, and 5, and the radicands are 2, 8, and 18.
First, multiply the coefficients: 3 × 2 × 5 = 30.
Next, multiply the radicands and simplify: √2 × √2 × √8 × √18 = √(2·2·8·18).
Now, simplify the radicands inside the square root: √(2·2·4·2·9·2) = √(2·2·2·2·4·2·9).
Since √2·2 = 2 and √4 = 2, we can simplify further: √(2·2·2·2·2) × √9 = √(2·2·2·2·4·4·9).
Extract the perfect squares: √(2·2) × √(2·2) × 2 × 2 × √9 = 2 × 2 × 2 × 2 × √9 = 16√3.
Finally, multiply this result by the coefficient: 30 × 16√3 = 480√3, which can be simplified to 60√3 by reducing the common factors of 30 and 16.
The correct answer is 60√3, which is choice D.