Which of the following is a like radical to $\sqrt[3]{6 x^2}$ ?
A. $x(\sqrt[3]{6 x})$
B. $6\left(\sqrt[3]{x^2}\right)$
C. $4\left(\sqrt[3]{6 x^2}\right)$
D. $x(\sqrt[3]{6})$
Answer (1)
Like radicals must have the same index and radicand.
Examine each option to see if it matches the form k 3 6 x 2 , where k is a constant or expression without radicals.
After analyzing all options, 4 ( 3 6 x 2 ) is the only like radical.
The like radical is 4 ( 3 6 x 2 ) .
Explanation
Understanding Like Radicals We are given the radical 3 6 x 2 and asked to identify which of the provided options is a like radical. Like radicals have the same index and radicand (the expression under the radical).
Identifying Key Properties A like radical to 3 6 x 2 must have an index of 3 and a radicand of 6 x 2 . We will examine each option to see if it matches this form.
Analyzing Option 1 Option 1: x ( 3 6 x ) = 3 6 x 3 × x = 3 6 x 4 . The radicand is 6 x 4 , which is not 6 x 2 . Thus, this is not a like radical.
Analyzing Option 2 Option 2: 6 ( 3 x 2 ) = 3 6 3 x 2 = 3 216 x 2 . The radicand is 216 x 2 , which is not 6 x 2 . Thus, this is not a like radical.
Analyzing Option 3 Option 3: 4 ( 3 6 x 2 ) . This has the same index (3) and radicand ( 6 x 2 ) as the original radical. Therefore, this is a like radical.
Analyzing Option 4 Option 4: x ( 3 6 ) = 3 x 3 × 6 = 3 6 x 3 . The radicand is 6 x 3 , which is not 6 x 2 . Thus, this is not a like radical.
Conclusion Therefore, the like radical is 4 ( 3 6 x 2 ) .
Examples
Like radicals are useful when simplifying expressions or solving equations involving radicals. For example, if you need to combine 2 3 6 x 2 + 4 3 6 x 2 , you can add the coefficients since the radicals are alike, resulting in 6 3 6 x 2 . This is similar to combining like terms in algebraic expressions.
