Write the expression
$2 x^2 y^4\left(\sqrt[4]{3 x^3 y}\right)$
as an entire radical.
Answer (1)
Raise the term outside the radical to the power of the index of the radical: ( 2 x 2 y 4 ) 4 = 16 x 8 y 16 .
Multiply the result with the expression inside the radical: 16 x 8 y 16 × 3 x 3 y = 48 x 11 y 17 .
Write the entire expression under the radical: 4 48 x 11 y 17 .
The expression as an entire radical is: 4 48 x 11 y 17
Explanation
Understanding the Problem We are given the expression 2 x^2 y^4\[ 4\sqrt{3 x^3 y}\right] and we want to write it as an entire radical. This means we want to move the term 2 x 2 y 4 inside the fourth root.
Raising to the Power of 4 To move 2 x 2 y 4 inside the fourth root, we need to raise it to the power of 4. So we have ( 2 x 2 y 4 ) 4 = 2 4 ( x 2 ) 4 ( y 4 ) 4 = 16 x 8 y 16 .
Multiplying Inside the Radical Now, we multiply this result with the expression inside the radical: 16 x 8 y 16 × 3 x 3 y = ( 16 × 3 ) ( x 8 × x 3 ) ( y 16 × y ) = 48 x 8 + 3 y 16 + 1 = 48 x 11 y 17 .
Writing as an Entire Radical Finally, we take the fourth root of the entire expression: 4 48 x 11 y 17 . Therefore, the expression 2 x 2 y 4 ( 4 3 x 3 y ) as an entire radical is 4 48 x 11 y 17 .
Examples
Understanding radicals is crucial in various fields, such as engineering and physics, where you might need to simplify complex expressions involving roots. For example, when calculating the period of a pendulum, the formula involves a square root. Simplifying such expressions makes calculations easier and more accurate. Radicals are also used in computer graphics for calculating distances and transformations.
