If `2x+(1)/(2x)=2`, then what is the value of `sqrt(2((1)/(x))^(4)+((1)/(x))^(5))`?

Options

A. 1
B. 2
C. 4
D. 8

Answer

To solve the equation given in the question, we will follow these steps:

### Step 1: Solve the equation \( 2x + \frac{1}{2x} = 2 \)

We start with the equation:
\[ 2x + \frac{1}{2x} = 2 \]

To eliminate the fraction, we can multiply both sides by \( 2x \):
\[ (2x)(2x) + (2x)\left(\frac{1}{2x}\right) = 2(2x) \]
This simplifies to:
\[ 4x^2 + 1 = 4x \]

### Step 2: Rearrange the equation

Next, we rearrange the equation to form a standard quadratic equation:
\[ 4x^2 - 4x + 1 = 0 \]

### Step 3: Factor the quadratic equation

We can factor the quadratic equation:
\[ (2x - 1)^2 = 0 \]

### Step 4: Solve for \( x \)

Setting the factor equal to zero gives us:
\[ 2x - 1 = 0 \implies 2x = 1 \implies x = \frac{1}{2} \]

### Step 5: Calculate \( \frac{1}{x} \)

Now that we have \( x \), we can find \( \frac{1}{x} \):
\[ \frac{1}{x} = \frac{1}{\frac{1}{2}} = 2 \]

### Step 6: Substitute \( \frac{1}{x} \) into the expression

We need to evaluate:
\[ \sqrt{2\left(\frac{1}{x}\right)^4 + \left(\frac{1}{x}\right)^5} \]
Substituting \( \frac{1}{x} = 2 \):
\[ \sqrt{2(2^4) + (2^5)} \]

### Step 7: Calculate \( 2^4 \) and \( 2^5 \)

Calculating the powers:
\[ 2^4 = 16 \quad \text{and} \quad 2^5 = 32 \]

### Step 8: Substitute back into the expression

Now substitute back:
\[ \sqrt{2(16) + 32} = \sqrt{32 + 32} = \sqrt{64} \]

### Step 9: Final calculation

Finally, we calculate:
\[ \sqrt{64} = 8 \]

Thus, the value is:
\[ \boxed{8} \]

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