Answer :
To determine which simplification correctly explains the statement
[tex]\(\sqrt[4]{2} = 2^{\frac{1}{4}}\)[/tex], let's break down the expression step-by-step.
Given the expression [tex]\(\left(2^{\frac{1}{4}}\right)^4\)[/tex]:
1. Expansion Step:
[tex]\[ \left(2^{\frac{1}{4}}\right)^4 = 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \][/tex]
2. Combining Exponents:
When multiplying powers of the same base, we add the exponents. Here, since each term has an exponent of [tex]\(\frac{1}{4}\)[/tex], we add the exponents together:
[tex]\[ 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} = 2^{\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}} \][/tex]
3. Simplify the Exponent:
Adding the exponents:
[tex]\[ \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = 1 \][/tex]
Thus, we have:
[tex]\[ 2^{\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}} = 2^1 \][/tex]
4. Evaluating the Power:
[tex]\[ 2^1 = 2 \][/tex]
Given the above steps, the correct simplification is present in option B:
```
B. [tex]\(\left(2^{\frac{1}{4}}\right)^4=2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}=2^{\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}}=2^1=2\)[/tex]
```
[tex]\(\sqrt[4]{2} = 2^{\frac{1}{4}}\)[/tex], let's break down the expression step-by-step.
Given the expression [tex]\(\left(2^{\frac{1}{4}}\right)^4\)[/tex]:
1. Expansion Step:
[tex]\[ \left(2^{\frac{1}{4}}\right)^4 = 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \][/tex]
2. Combining Exponents:
When multiplying powers of the same base, we add the exponents. Here, since each term has an exponent of [tex]\(\frac{1}{4}\)[/tex], we add the exponents together:
[tex]\[ 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} = 2^{\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}} \][/tex]
3. Simplify the Exponent:
Adding the exponents:
[tex]\[ \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = 1 \][/tex]
Thus, we have:
[tex]\[ 2^{\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}} = 2^1 \][/tex]
4. Evaluating the Power:
[tex]\[ 2^1 = 2 \][/tex]
Given the above steps, the correct simplification is present in option B:
```
B. [tex]\(\left(2^{\frac{1}{4}}\right)^4=2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}=2^{\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}}=2^1=2\)[/tex]
```
