To solve the given problem, we need to simplify each of the given radical expressions and then see if any of them can be combined.
First, let's list out the expressions we need to simplify:
1. [tex]\(3 \sqrt{7}\)[/tex]
2. [tex]\( -5 \sqrt[4]{7} \)[/tex]
3. [tex]\( -2 \sqrt{7} \)[/tex]
4. [tex]\( -2 \sqrt[3]{7} \)[/tex]
### Simplifying Each Expression:
1. [tex]\( 3 \sqrt{7} \)[/tex]:
This expression is already simplified. We have [tex]\( 3 \)[/tex] times the square root of [tex]\( 7 \)[/tex]:
[tex]\[
3 \sqrt{7} \approx 3 \cdot 2.6457513110645907 \approx 7.937253933193772
\][/tex]
2. [tex]\( -5 \sqrt[4]{7} \)[/tex]:
This expression is also simplified. We have [tex]\(-5\)[/tex] times the fourth root of [tex]\( 7 \)[/tex]:
[tex]\[
-5 \sqrt[4]{7} \approx -5 \cdot 1.6266412073914813 \approx -8.132882808488928
\][/tex]
3. [tex]\( -2 \sqrt{7} \)[/tex]:
This one is straightforward as well. We have [tex]\( -2 \)[/tex] times the square root of [tex]\( 7 \)[/tex]:
[tex]\[
-2 \sqrt{7} \approx -2 \cdot 2.6457513110645907 \approx -5.291502622129181
\][/tex]
4. [tex]\( -2 \sqrt[3]{7} \)[/tex]:
We have [tex]\(-2\)[/tex] times the cube root of [tex]\( 7 \)[/tex]:
[tex]\[
-2 \sqrt[3]{7} \approx -2 \cdot 1.912931182772389 \approx -3.825862365544778
\][/tex]
### Summing Terms That Can Be Simplified:
Next, we look at terms that can be combined. Notice we have two terms that involve [tex]\(\sqrt{7}\)[/tex]:
- [tex]\(3 \sqrt{7}\)[/tex] and [tex]\(-2 \sqrt{7}\)[/tex]
We add these coefficients:
[tex]\[
3 \sqrt{7} + (-2 \sqrt{7}) = (3 - 2) \sqrt{7} = 1 \sqrt{7} \approx 2.6457513110645907
\][/tex]
### Final Answer:
Combining all terms and their simplified forms, we have:
[tex]\[
3 \sqrt{7} - 5 \sqrt[4]{7} - 2 \sqrt{7} - 2 \sqrt[3]{7}
\][/tex]
The simplified terms, in terms of approximate values, are:
1. [tex]\(3 \sqrt{7} \approx 7.937253933193772\)[/tex]
2. [tex]\(-5 \sqrt[4]{7} \approx -8.132882808488928\)[/tex]
3. [tex]\(-2 \sqrt{7} \approx -5.291502622129181\)[/tex]
4. [tex]\(-2 \sqrt[3]{7} \approx -3.825862365544778\)[/tex]
The sum of the simplified like terms is approximately:
[tex]\[
1 \sqrt{7} \approx 2.6457513110645907
\][/tex]
This concludes our step-by-step simplification.
