Multiply.
$4 \sqrt{2} \times 6 \sqrt{18}$
A. $20 \sqrt{5}$
B. $144 \sqrt{2}$
C. 144
D. $24 \sqrt{6}$
Answer (1)
Multiply the coefficients: 4 × 6 = 24 .
Multiply the terms inside the square roots: 2 × 18 = 36 .
Simplify the square root: 36 = 6 .
Multiply the results: 24 × 6 = 144 . The final answer is 144 .
Explanation
Understanding the problem We are given the expression 4 2 × 6 18 to simplify. Our goal is to multiply these terms and express the result in its simplest form.
Multiplying Coefficients First, let's multiply the coefficients (the numbers outside the square roots):
4 × 6 = 24
Multiplying Square Roots Next, let's multiply the terms inside the square roots:
2 × 18 = 2 × 18 = 36
Simplifying the Square Root Now, we simplify the square root:
36 = 6
Final Calculation Finally, we multiply the result from the coefficients and the simplified square root:
24 × 6 = 144
So, 4 2 × 6 18 = 144
Examples
Understanding how to simplify radical expressions is useful in various fields, such as engineering and physics, where calculations involving lengths and distances are common. For example, when calculating the length of a diagonal in a square or the distance between two points, you often encounter square roots. Simplifying these expressions makes further calculations easier and more accurate. Imagine you are building a square garden and need to determine the length of a support beam that will run diagonally across the garden. The diagonal length might initially be expressed as 4 2 × 6 18 feet, but by simplifying it to 144 feet, you immediately know the required length of the beam.
