Question
R.1 Roots of integers EUH Find all real fourth roots of 256. If there are multiple answers, separate them with commas. Write your answers in simplified form.
Ask by Bolton Barrett. in Mexico
Mar 03,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The real fourth roots of 256 are 4 and -4.
Solution
To find all real fourth roots of 256, you need to solve the equation x⁴ = 256.
Since 256 can be expressed as 4⁴ (because 4⁴ = 256), the equation becomes:
x⁴ = 4⁴
Taking the fourth root of both sides gives two real solutions:
x = 4 and x = -4
Thus, the real fourth roots of 256, written in simplified form, are:
4, -4
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Simplify this solution Bonus Knowledge
The fourth roots of 256 can be found by realizing that \(256 = 2^8\). To find the fourth root, you can divide the exponent by 4. This gives you: \[ \sqrt[4]{256} = \sqrt[4]{2^8} = 2^{8/4} = 2^2 = 4. \] However, since we are looking for all real fourth roots, we must remember that fourth roots can be positive or negative for even roots. Therefore, the real fourth roots of 256 are: \[ 4, -4. \]
