Sure, I can help you simplify each of the given radical expressions step by step.
### Expression 1: [tex]\( 3 \sqrt{7} - 5 \sqrt[4]{7} \)[/tex]
We start with the expression [tex]\( 3 \sqrt{7} - 5 \sqrt[4]{7} \)[/tex].
Here, [tex]\( \sqrt{7} \)[/tex] represents [tex]\( 7^{1/2} \)[/tex], and [tex]\( \sqrt[4]{7} \)[/tex] represents [tex]\( 7^{1/4} \)[/tex].
So, rewriting the expression we get:
[tex]\[ 3 \sqrt{7} - 5 \sqrt[4]{7} \][/tex]
Which translates to:
[tex]\[ 3 \cdot 7^{1/2} - 5 \cdot 7^{1/4} \][/tex]
This expression cannot be simplified further due to the unlike radicals involving different root powers.
Thus, the simplified form of this expression remains:
[tex]\[ 3 \sqrt{7} - 5 \sqrt[4]{7} \][/tex]
### Expression 2: [tex]\( -2 \sqrt{7} \)[/tex]
Here, [tex]\( \sqrt{7} \)[/tex] represents [tex]\( 7^{1/2} \)[/tex].
So, rewriting the expression we get:
[tex]\[ -2 \sqrt{7} \][/tex]
Which translates to:
[tex]\[ -2 \cdot 7^{1/2} \][/tex]
This is in its simplest form and does not require further simplification.
Thus, the simplified form of this expression remains:
[tex]\[ -2 \sqrt{7} \][/tex]
### Expression 3: [tex]\( -2 \sqrt[3]{7} \)[/tex]
Here, [tex]\( \sqrt[3]{7} \)[/tex] represents [tex]\( 7^{1/3} \)[/tex].
So, rewriting the expression we get:
[tex]\[ -2 \sqrt[3]{7} \][/tex]
Which translates to:
[tex]\[ -2 \cdot 7^{1/3} \][/tex]
This is also in its simplest form and does not require further simplification.
Thus, the simplified form of this expression remains:
[tex]\[ -2 \sqrt[3]{7} \][/tex]
### Final Simplified Expressions
Summarizing our results, the simplified forms of the given expressions are:
1. [tex]\( 3 \sqrt{7} - 5 \sqrt[4]{7} \)[/tex]
2. [tex]\( -2 \sqrt{7} \)[/tex]
3. [tex]\( -2 \sqrt[3]{7} \)[/tex]
In a more mathematical representation with exact fractional exponents, they are:
1. [tex]\( -5 \cdot 7^{1/4} + 3 \sqrt{7} \)[/tex]
2. [tex]\( -2 \sqrt{7} \)[/tex]
3. [tex]\( -2 \cdot 7^{1/3} \)[/tex]
