Simplify the root.
${}^4\sqrt{256}$

• A. $64$
• B. $16$
• C. $4$
• D. $8$

Answer: (C)

Verified step by step guidance

1. Understand the problem: You need to simplify the fourth root of the square root of 256, which can be written as \(\sqrt[4]{\sqrt{256}}\).
2. Rewrite the expression using fractional exponents: Recall that \(\sqrt{256} = 256^{\frac{1}{2}}\), so the entire expression becomes \(\left(256^{\frac{1}{2}}\right)^{\frac{1}{4}}\).
3. Apply the power of a power rule: When raising a power to another power, multiply the exponents. So, \(\left(256^{\frac{1}{2}}\right)^{\frac{1}{4}} = 256^{\frac{1}{2} \times \frac{1}{4}} = 256^{\frac{1}{8}}\).
4. Simplify the base if possible: Since 256 is a power of 2 (because \$256 = 2^8$), rewrite the expression as \(\left(2^8\right)^{\frac{1}{8}}\).
5. Apply the power of a power rule again: Multiply the exponents to get \(2^{8 \times \frac{1}{8}} = 2^1\), which simplifies to 2.