Solve for x . −x4−2=−10

Answer 1

To solve the equation \(-x^4 - 2 = -10\), we can start by isolating \(x^4\).

Step 1: Add 2 to both sides.

\[ -x^4 - 2 + 2 = -10 + 2 \]

This simplifies to:

\[ -x^4 = -8 \]

Step 2: Multiply both sides by -1.

\[ x^4 = 8 \]

Step 3: Take the fourth root of both sides.

\[ x = \pm \sqrt[4]{8} \]

Step 4: Simplify \(\sqrt[4]{8}\).

We can express 8 as \(2^3\):

\[ \sqrt[4]{8} = \sqrt[4]{2^3} = 2^{3/4} = 2^{0.75} = 2^{\frac{3}{4}} = \sqrt{2^3} / 2^{1/4} = \frac{\sqrt{8}}{2^{1/4}} = \frac{2\sqrt{2}}{\sqrt[4]{2}} \]

Thus, the solutions for \(x\) are:

\[ x = \pm \sqrt[4]{8} \approx \pm 1.68179 \]

So, the final answer is:

\[ x = \pm 2^{3/4} \] or approximately \(x \approx \pm 1.68179\).

Answer 2

That is wrong

Answer 3

Let’s resolve the equation step by step correctly this time:

Starting with the original equation:

\[ -x^4 - 2 = -10 \]

Step 1: Add 2 to both sides.

\[ -x^4 - 2 + 2 = -10 + 2 \]

This simplifies to:

\[ -x^4 = -8 \]

Step 2: Multiply both sides by -1.

\[ x^4 = 8 \]

Step 3: Take the fourth root of both sides.

\[ x = \pm \sqrt[4]{8} \]

Step 4: Simplify \(\sqrt[4]{8}\).

We know that \(8 = 2^3\), so:

\[ x = \pm \sqrt[4]{2^3} = \pm 2^{3/4} \]

This can also be expressed more simply if necessary:

\[ 2^{3/4} = \sqrt{2^3} = \sqrt{8} \div \sqrt[4]{16} = 2 \sqrt[4]{2} \]

Thus, the solutions for \(x\) are:

\[ x = \pm \sqrt[4]{8} \quad \text{or} \quad x = \pm 2^{3/4} \]

To get a numerical approximation for the positive root:

\[ x \approx \pm 1.68179 \]

Thank you for your patience. Please let me know if you need further clarification or assistance!

Answer 4

That is still wrong

Answer 5

I apologize for the confusion in my previous responses. Let's solve the equation \( -x^4 - 2 = -10 \) step by step correctly.

Given:

\[ -x^4 - 2 = -10 \]

Step 1: Add 2 to both sides.

\[ -x^4 - 2 + 2 = -10 + 2 \]

This simplifies to:

\[ -x^4 = -8 \]

Step 2: Multiply both sides by -1.

\[ x^4 = 8 \]

Step 3: Solve for \(x\).

Now take the fourth root of both sides:

\[ x = \pm \sqrt[4]{8} \]

Using the fact that \(8\) can be expressed as \(2^3\):

\[ x = \pm \sqrt[4]{2^3} = \pm 2^{3/4} \]

To convert \(2^{3/4}\) back to a decimal:

\[ 2^{3/4} = (2^{0.75}) \approx 1.68179 \]

Thus, the complete solution for \(x\) is:

\[ x = \pm \sqrt[4]{8} \quad \text{or} \quad x \approx \pm 1.68179 \]

If you believe this approach is incorrect, please guide me on what specifically you think should be done differently, and I appreciate your patience!