To determine the like radicals from the given list, we must compare both the radicands (the expressions under the radical) and the indices (the roots' degrees) to ensure they are the same.
Let's analyze each expression step by step:
1. [tex]\( 9 \sqrt{6 a b^2} \)[/tex]
- Radicand: [tex]\( 6ab^2 \)[/tex]
- Index: 2 (square root is a root of degree 2)
2. [tex]\( -5 \sqrt{6 a b^2} \)[/tex]
- Radicand: [tex]\( 6ab^2 \)[/tex]
- Index: 2 (square root is a root of degree 2)
3. [tex]\( 4 \sqrt[3]{6 a b^2} \)[/tex]
- Radicand: [tex]\( 6ab^2 \)[/tex]
- Index: 3 (cube root is a root of degree 3)
4. [tex]\( 2 \sqrt{6 b^2 a} \)[/tex]
- Radicand: [tex]\( 6b^2a \)[/tex]. Note that [tex]\( 6b^2a \)[/tex] is the same as [tex]\( 6ab^2 \)[/tex], just written in a different order due to the commutative property of multiplication.
- Index: 2 (square root is a root of degree 2)
Comparing the expressions:
- The first expression, [tex]\( 9 \sqrt{6 a b^2} \)[/tex], has a radicand of [tex]\( 6ab^2 \)[/tex] and an index of 2.
- The second expression, [tex]\( -5 \sqrt{6 a b^2} \)[/tex], has the same radicand, [tex]\( 6ab^2 \)[/tex], and the same index, 2. These two are like radicals.
- The third expression, [tex]\( 4 \sqrt[3]{6 a b^2} \)[/tex], has the same radicand, [tex]\( 6ab^2 \)[/tex], but a different index, 3. It is not a like radical with the first two.
- The fourth expression, [tex]\( 2 \sqrt{6 b^2 a} \)[/tex], has the radicand [tex]\( 6b^2a \)[/tex] which is the same as [tex]\( 6ab^2 \)[/tex], and the same index, 2. This means it is a like radical with the first two.
Therefore, the like radicals in the given list are:
- [tex]\( 9 \sqrt{6 a b^2} \)[/tex]
- [tex]\( -5 \sqrt{6 a b^2} \)[/tex]
- [tex]\( 2 \sqrt{6 b^2 a} \)[/tex]
So, the indices of the like radicals are: 0, 1, and 3.
