If `x^(3)+(1)/(x^(3))=62`, then find the value of `sqrt(x^(3))+(1)/(sqrt(x^(3)))`.
The following steps are involved in solving the above problem. Arrange them in sequential order.
(A) `therefore(sqrt(x^(3))+(1)/(sqrt(x^(3))))^(2)=62+2=64`
(B) `implies(sqrt(x^(3))+(1)/(sqrt(x^(3))))=sqrt(64)=8`
(C ) `therefore(sqrt(x^(3))+(1)/(sqrt(x^(3))))^(2)=x^(3)+(1)/(x^(3))+2`
(D) But `x^(3)+(1)/(x^(3))=62`

Options

A. CABD
B. ABCD
C. ACDB
D. CDAB

Answer

To solve the problem step by step, we start with the equation given in the question:

1. **Given:**
\( x^3 + \frac{1}{x^3} = 62 \)

2. **We need to find:**
\( \sqrt{x^3} + \frac{1}{\sqrt{x^3}} \)

3. **Using the identity:**
We know that:
\[ \left( \sqrt{x^3} + \frac{1}{\sqrt{x^3}} \right)^2 = x^3 + \frac{1}{x^3} + 2 \]
This is derived from the expansion of \( (a + b)^2 = a^2 + b^2 + 2ab \).

4. **Substituting the known value:**
From the given information, we can substitute \( x^3 + \frac{1}{x^3} = 62 \) into the identity:
\[ \left( \sqrt{x^3} + \frac{1}{\sqrt{x^3}} \right)^2 = 62 + 2 \]
\[ \left( \sqrt{x^3} + \frac{1}{\sqrt{x^3}} \right)^2 = 64 \]

5. **Taking the square root:**
Now, we take the square root of both sides:
\[ \sqrt{x^3} + \frac{1}{\sqrt{x^3}} = \sqrt{64} \]
\[ \sqrt{x^3} + \frac{1}{\sqrt{x^3}} = 8 \]

Thus, the final answer is:
**The value of \( \sqrt{x^3} + \frac{1}{\sqrt{x^3}} \) is 8.**

### Sequence of Steps:
1. (C) \( \left( \sqrt{x^3} + \frac{1}{\sqrt{x^3}} \right)^2 = x^3 + \frac{1}{x^3} + 2 \)
2. (D) But \( x^3 + \frac{1}{x^3} = 62 \)
3. (A) Therefore \( \left( \sqrt{x^3} + \frac{1}{\sqrt{x^3}} \right)^2 = 62 + 2 = 64 \)
4. (B) Implies \( \sqrt{x^3} + \frac{1}{\sqrt{x^3}} = \sqrt{64} = 8 \)