Simplify each radical expression, if possible.

[tex]\[ 3 \sqrt{7} - 5 \sqrt[4]{7} \][/tex]

A. [tex]\(-2 \sqrt{7}\)[/tex]

B. [tex]\(-2 \sqrt[3]{7}\)[/tex]

C. Cannot be simplified



Answer :

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]