Select the simplification that accurately explains the following statement. $$ \sqrt[2]{7}=7 \frac{1}{3} $$ A. $$ \left(7^{\frac{1}{2}}\right)^{3}=7^{\frac{1}{4}} \cdot 7^{\frac{1}{1}} \cdot 7^{\frac{1}{1}}=7^{\frac{1}{3}+\frac{1}{1}}=7^{1}=7^{1}=7 $$ B. $$ \left(7^{\frac{1}{4}}\right)^{3}=7^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot 7^{\frac{1}{3}}=7 \cdot\left(\frac{1}{3}+\frac{1}{3}+\frac{1}{3}\right)-7 \cdot \frac{3}{3}=7 \cdot 1=7 $$ c. $$ \left(7^{\frac{1}{2}}\right)^{3}=7^{\frac{1}{2}} \cdot 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}}=7 \cdot 7^{\frac{1}{1}}=3 \cdot \frac{1}{3} \cdot 7=1 \cdot 7=7 $$ D. $$ \left(7^{\frac{1}{3}}\right)^{3}=7^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot 7^{\frac{1}{3}}=7^{\frac{1}{3}} \cdot \frac{1}{4} \cdot \frac{1}{4}=7^{\frac{1}{1}}=7^{1}=7 $$

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UpStudy AI · Accepted Answer

Upon examining the statement and the provided simplification options, it's clear that there are inconsistencies and errors in both the original statement and the simplifications. Let's break it down:

### Original Statement:
$$
\sqrt[2]{7} = 7 \frac{1}{3}
$$
This can be interpreted as:
$$
\sqrt{7} = 7 \times \frac{1}{3} = \frac{7}{3}
$$
However, $$\sqrt{7} \approx 2.6458$$ and $$\frac{7}{3} \approx 2.3333$$, so the equality does not hold true.

### Evaluating the Simplifications:

**Option A:**
$$
\left(7^{\frac{1}{2}}\right)^{3} = 7^{\frac{3}{2}} = 7^{\frac{1}{4} + 1 + 1} = 7^{2.25} \neq 7
$$
**Issue:** The exponent addition is incorrect. $$ \frac{1}{2} \times 3 = \frac{3}{2} $$, not $$ \frac{1}{4} + 1 + 1 $$.

---

**Option B:**
$$
\left(7^{\frac{1}{4}}\right)^{3} = 7^{\frac{3}{4}} = 7 \times \left(\frac{1}{3} + \frac{1}{3} + \frac{1}{3}\right) - 7 \times \frac{3}{3} = 7 \times 1 = 7
$$
**Issue:** The manipulation of exponents is incorrect. $$ \frac{1}{4} \times 3 = \frac{3}{4} $$, and the subsequent steps don't logically follow from exponent rules.

---

**Option C:**
$$
\left(7^{\frac{1}{2}}\right)^{3} = 7^{\frac{3}{2}} = 7 \times 7^{1} = 3 \times \frac{1}{3} \times 7 = 1 \times 7 = 7
$$
**Issue:** The steps involve incorrect exponent manipulation and arbitrary multiplication factors that do not adhere to mathematical principles.

---

**Option D:**
$$
\left(7^{\frac{1}{3}}\right)^{3} = 7^{1} = 7^{\frac{1}{4}} \times 7^{\frac{1}{4}} \times 7^{\frac{1}{3}} = 7^{\frac{1}{3}} \times \frac{1}{4} \times \frac{1}{4} = 7^{1} = 7^{1} = 7
$$
**Issue:** Similar to the previous options, this simplification incorrectly applies exponent rules and introduces unjustified multiplication factors.

### Conclusion:

**None of the provided options (A, B, C, or D) correctly simplify the original statement.** Additionally, the original statement itself is mathematically incorrect because $$\sqrt{7} \neq \frac{7}{3}$$.

### Correct Approach:

If the goal is to simplify expressions involving exponents, it's crucial to apply the exponent rules accurately. For example:

- $$\left(7^{\frac{1}{2}}\right)^3 = 7^{\frac{3}{2}} = 7^{1} \times 7^{\frac{1}{2}} = 7 \sqrt{7}$$
- $$\left(7^{\frac{1}{3}}\right)^3 = 7^{1} = 7$$

These simplifications adhere to the rules of exponents without introducing incorrect operations.
None of the provided options correctly simplify the original statement, and the original statement itself is mathematically incorrect because $$\sqrt{7} \neq \frac{7}{3}$$.

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