If $x + \frac{1}{x}$ = 4 , then the value of $ x^{4} + \frac{1}{x^{4}}$ is:

Q: If $x + \frac{1}{x}$ = 4 , then the value of $ x^{4} + \frac{1}{x^{4}}$ is:

To solve for the value of $ x^4 + \frac{1}{x^4} $ given that $ x + \frac{1}{x} = 4 $, we can use algebraic identities and manipulation. 1. **Step 1: Find $ x^2 + \frac{1}{x^2} $** We start with the given equation: \[ x + \frac{1}{x} = 4 \] Square both sides to find $ x^2 + \frac{1}{x^2} $: \[ \left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2} \] \[ 4^2 = x^2 + 2 + \frac{1}{x^2} \] \[ 16 = x^2 + \frac{1}{x^2} + 2 \] \[ x^2 + \frac{1}{x^2} = 16 - 2 = 14 \] 2. **Step 2: Find $ x^4 + \frac{1}{x^4} $** Now, square $ x^2 + \frac{1}{x^2} $ to find $ x^4 + \frac{1}{x^4} $: \[ \left(x^2 + \frac{1}{x^2}\right)^2 = x^4 + 2 + \frac{1}{x^4} \] \[ 14^2 = x^4 + 2 + \frac{1}{x^4} \] \[ 196 = x^4 + \frac{1}{x^4} + 2 \] \[ x^4 + \frac{1}{x^4} = 196 - 2 = 194 \] Thus, the value of $ x^4 + \frac{1}{x^4} $ is 194, which corresponds to Option 1. **Explanation for Incorrect Options:** - **Option 2 (174):** This value is incorrect because, as shown in the calculations, the correct value derived from the given equation is 194, not 174. - **Option 3 (144):** This value is incorrect because the calculations clearly show that the result of $ x^4 + \frac{1}{x^4} $ is 194, not 144. - **Option 4 (154):** This value is incorrect because the step-by-step algebraic manipulation confirms that the correct value is 194, not 154. Therefore, Option 1 is the correct answer, as it accurately reflects the calculated value of $ x^4 + \frac{1}{x^4} $.

Question

Accepted Answer

To solve for the value of $ x^4 + \frac{1}{x^4} $ given that $ x + \frac{1}{x} = 4 $, we can use algebraic identities and manipulation.

1. **Step 1: Find $ x^2 + \frac{1}{x^2} $**

We start with the given equation:
   $$
   x + \frac{1}{x} = 4
   $$

Square both sides to find $ x^2 + \frac{1}{x^2} $:
   $$
   \left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2}
   $$
   $$
   4^2 = x^2 + 2 + \frac{1}{x^2}
   $$
   $$
   16 = x^2 + \frac{1}{x^2} + 2
   $$
   $$
   x^2 + \frac{1}{x^2} = 16 - 2 = 14
   $$

2. **Step 2: Find $ x^4 + \frac{1}{x^4} $**

Now, square $ x^2 + \frac{1}{x^2} $ to find $ x^4 + \frac{1}{x^4} $:
   $$
   \left(x^2 + \frac{1}{x^2}\right)^2 = x^4 + 2 + \frac{1}{x^4}
   $$
   $$
   14^2 = x^4 + 2 + \frac{1}{x^4}
   $$
   $$
   196 = x^4 + \frac{1}{x^4} + 2
   $$
   $$
   x^4 + \frac{1}{x^4} = 196 - 2 = 194
   $$

Thus, the value of $ x^4 + \frac{1}{x^4} $ is 194, which corresponds to Option 1.

**Explanation for Incorrect Options:**

- **Option 2 (174):** This value is incorrect because, as shown in the calculations, the correct value derived from the given equation is 194, not 174.

- **Option 3 (144):** This value is incorrect because the calculations clearly show that the result of $ x^4 + \frac{1}{x^4} $ is 194, not 144.

- **Option 4 (154):** This value is incorrect because the step-by-step algebraic manipulation confirms that the correct value is 194, not 154.

Therefore, Option 1 is the correct answer, as it accurately reflects the calculated value of $ x^4 + \frac{1}{x^4} $.