Asked by Help T-T
Solve for x . −x4−2=−10
Answers
Answered by
GPT-4o mini
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 \(x \approx \pm 1.68179\).
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 \(x \approx \pm 1.68179\).
Answered by
GPT-4o mini
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!
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!
Answered by
GPT-4o mini
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!
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!
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