Let's simplify each of the given radical expressions step-by-step.
1. Expression 1: [tex]\( 3 \sqrt{7} \)[/tex]
- The term [tex]\( \sqrt{7} \)[/tex] represents the square root of 7.
- Multiplying [tex]\( 3 \)[/tex] by [tex]\( \sqrt{7} \)[/tex] gives [tex]\( 3 \sqrt{7} \)[/tex].
- This expression is already in its simplest form since [tex]\( \sqrt{7} \)[/tex] cannot be simplified further.
- The simplified expression is [tex]\( 3 \sqrt{7} \)[/tex].
2. Expression 2: [tex]\( -5 \sqrt[4]{7} \)[/tex]
- The term [tex]\( \sqrt[4]{7} \)[/tex] represents the fourth root of 7, which is also denoted as [tex]\( 7^{1/4} \)[/tex].
- Multiplying [tex]\( -5 \)[/tex] by [tex]\( \sqrt[4]{7} \)[/tex] results in [tex]\( -5 \sqrt[4]{7} \)[/tex].
- This expression is already in its simplest form since [tex]\( \sqrt[4]{7} = 7^{1/4} \)[/tex] cannot be simplified further.
- The simplified expression is [tex]\( -5 \sqrt[4]{7} \)[/tex], which can also be written as [tex]\( -5 \cdot 7^{1/4} \)[/tex].
3. Expression 3: [tex]\( -2 \sqrt{7} \)[/tex]
- The term [tex]\( \sqrt{7} \)[/tex] represents the square root of 7.
- Multiplying [tex]\( -2 \)[/tex] by [tex]\( \sqrt{7} \)[/tex] gives [tex]\( -2 \sqrt{7} \)[/tex].
- This expression is already in its simplest form since [tex]\( \sqrt{7} \)[/tex] cannot be simplified further.
- The simplified expression is [tex]\( -2 \sqrt{7} \)[/tex].
4. Expression 4: [tex]\( -2 \sqrt[3]{7} \)[/tex]
- The term [tex]\( \sqrt[3]{7} \)[/tex] represents the cube root of 7, which is also denoted as [tex]\( 7^{1/3} \)[/tex].
- Multiplying [tex]\( -2 \)[/tex] by [tex]\( \sqrt[3]{7} \)[/tex] results in [tex]\( -2 \sqrt[3]{7} \)[/tex].
- This expression is already in its simplest form since [tex]\( \sqrt[3]{7} = 7^{1/3} \)[/tex] cannot be simplified further.
- The simplified expression is [tex]\( -2 \sqrt[3]{7} \)[/tex], which can also be written as [tex]\( -2 \cdot 7^{1/3} \)[/tex].
So, the simplified versions of the given expressions are:
- [tex]\( 3 \sqrt{7} \)[/tex]
- [tex]\( -5 \sqrt[4]{7} \)[/tex] or [tex]\( -5 \cdot 7^{1/4} \)[/tex]
- [tex]\( -2 \sqrt{7} \)[/tex]
- [tex]\( -2 \sqrt[3]{7} \)[/tex] or [tex]\( -2 \cdot 7^{1/3} \)[/tex]
