Let's begin by understanding the concept of like radicals. Like radicals have the same radicands (the expression under the radical symbol) and the same type of root (square root, cube root, etc.).
Given the expressions:
1. [tex]\( 9 \sqrt{6ab^2} \)[/tex]
2. [tex]\( -5 \sqrt{6ab^2} \)[/tex]
3. [tex]\( 4 \sqrt[3]{6ab^2} \)[/tex]
4. [tex]\( 2 \sqrt{6b^2a} \)[/tex]
We will first convert and compare the radicands of each radical to determine if they are like radicals.
1. [tex]\( 9 \sqrt{6ab^2} \)[/tex]
- The radicand is [tex]\( 6ab^2 \)[/tex].
- The type of root is a square root.
2. [tex]\( -5 \sqrt{6ab^2} \)[/tex]
- The radicand is [tex]\( 6ab^2 \)[/tex].
- The type of root is a square root.
3. [tex]\( 4 \sqrt[3]{6ab^2} \)[/tex]
- The radicand is [tex]\( 6ab^2 \)[/tex].
- The type of root is a cube root (not a square root).
4. [tex]\( 2 \sqrt{6b^2a} \)[/tex]
- The radicand is [tex]\( 6b^2a \)[/tex], which is the same as [tex]\( 6ab^2 \)[/tex].
- The type of root is a square root.
Now, let's determine which of these are like radicals. A like radical pair would require both the radicand and the type of root to be identical.
- The radicand in expressions 1, 2, and 4 is [tex]\( 6ab^2 \)[/tex] and the type of root is a square root.
- The radicand in expression 3 is [tex]\( 6ab^2 \)[/tex] but the type of root is a cube root, which makes it different from the others.
Therefore, the radicals that are similar are:
1. [tex]\( 9 \sqrt{6ab^2} \)[/tex]
2. [tex]\( -5 \sqrt{6ab^2} \)[/tex]
4. [tex]\( 2 \sqrt{6ab^2} \)[/tex]
So, the like radicals are:
1. [tex]\( 9 \sqrt{6ab^2} \)[/tex]
2. [tex]\( -5 \sqrt{6ab^2} \)[/tex]
4. [tex]\( 2 \sqrt{6ab^2} \)[/tex]
In conclusion, the expressions that are like radicals are:
[tex]\[ 9 \sqrt{6ab^2}, -5 \sqrt{6ab^2}, \text{and} 2 \sqrt{6ab^2} \][/tex]
