Answer :
To determine which expressions are in their simplest form, we need to simplify each expression and compare it with the given form. Let's go through them one by one.
1. Expression: [tex]\( 5 \sqrt{3b} \)[/tex]
Simplified Form: [tex]\( 5 \sqrt{3} \sqrt{b} \)[/tex]
The expression is not simplified because it can be written as a product of separate square roots.
2. Expression: [tex]\( 2 \sqrt{21} \)[/tex]
Simplified Form: [tex]\( 2 \sqrt{21} \)[/tex]
This expression is already in its simplest form since [tex]\( 21 \)[/tex] is not a perfect square, and there are no further simplifications possible.
3. Expression: [tex]\( x \sqrt{8} \)[/tex]
Simplified Form: [tex]\( x \sqrt{4 \cdot 2} = x \cdot 2 \sqrt{2} = 2x \sqrt{2} \)[/tex]
The initial form [tex]\( x \sqrt{8} \)[/tex] can be simplified, hence it is not in its simplest form.
4. Expression: [tex]\( 2 y \sqrt{36} \)[/tex]
Simplified Form: [tex]\( 2 y \sqrt{36} = 2 y \cdot 6 = 12 y \)[/tex]
The original form [tex]\( 2 y \sqrt{36} \)[/tex] simplifies to [tex]\( 12 y \)[/tex], so it is not in its simplest form.
5. Expression: [tex]\( \sqrt{5} \)[/tex]
Simplified Form: [tex]\( \sqrt{5} \)[/tex]
This expression is already in its simplest form since [tex]\( 5 \)[/tex] is not a perfect square.
6. Expression: [tex]\( c \sqrt{12 c^2} \)[/tex]
Simplified Form: [tex]\( c \sqrt{12 c^2} = c \cdot \sqrt{4 \cdot 3 \cdot c^2} = c \cdot 2 \sqrt{3} \cdot c = 2c^2 \sqrt{3} \)[/tex]
The initial form [tex]\( c \sqrt{12 c^2} \)[/tex] simplifies to [tex]\( 2 \sqrt{3} c \cdot c \)[/tex], so it is not in its simplest form.
Based on the detailed step-by-step simplifications, the expressions that are listed in the simplest form are:
1. [tex]\( 2 \sqrt{21} \)[/tex]
2. [tex]\( \sqrt{5} \)[/tex]
These are the expressions that cannot be simplified any further and are already in their simplest forms.
1. Expression: [tex]\( 5 \sqrt{3b} \)[/tex]
Simplified Form: [tex]\( 5 \sqrt{3} \sqrt{b} \)[/tex]
The expression is not simplified because it can be written as a product of separate square roots.
2. Expression: [tex]\( 2 \sqrt{21} \)[/tex]
Simplified Form: [tex]\( 2 \sqrt{21} \)[/tex]
This expression is already in its simplest form since [tex]\( 21 \)[/tex] is not a perfect square, and there are no further simplifications possible.
3. Expression: [tex]\( x \sqrt{8} \)[/tex]
Simplified Form: [tex]\( x \sqrt{4 \cdot 2} = x \cdot 2 \sqrt{2} = 2x \sqrt{2} \)[/tex]
The initial form [tex]\( x \sqrt{8} \)[/tex] can be simplified, hence it is not in its simplest form.
4. Expression: [tex]\( 2 y \sqrt{36} \)[/tex]
Simplified Form: [tex]\( 2 y \sqrt{36} = 2 y \cdot 6 = 12 y \)[/tex]
The original form [tex]\( 2 y \sqrt{36} \)[/tex] simplifies to [tex]\( 12 y \)[/tex], so it is not in its simplest form.
5. Expression: [tex]\( \sqrt{5} \)[/tex]
Simplified Form: [tex]\( \sqrt{5} \)[/tex]
This expression is already in its simplest form since [tex]\( 5 \)[/tex] is not a perfect square.
6. Expression: [tex]\( c \sqrt{12 c^2} \)[/tex]
Simplified Form: [tex]\( c \sqrt{12 c^2} = c \cdot \sqrt{4 \cdot 3 \cdot c^2} = c \cdot 2 \sqrt{3} \cdot c = 2c^2 \sqrt{3} \)[/tex]
The initial form [tex]\( c \sqrt{12 c^2} \)[/tex] simplifies to [tex]\( 2 \sqrt{3} c \cdot c \)[/tex], so it is not in its simplest form.
Based on the detailed step-by-step simplifications, the expressions that are listed in the simplest form are:
1. [tex]\( 2 \sqrt{21} \)[/tex]
2. [tex]\( \sqrt{5} \)[/tex]
These are the expressions that cannot be simplified any further and are already in their simplest forms.
