Simplify the following.
${}^4\sqrt{256a^8{b}^9}$

• A. $4a^2{b}^3$
• B. $4a^2b^2.^4\(\sqrt{b}\)$
• C. $4a^4b^5$
• D. $4a^2b^2.^4\(\sqrt{b^3}\)$

Answer: (B)

Verified step by step guidance

1. Recognize that the expression is a fourth root of the product \$256a^8b^9$, written as \(^4\sqrt{256a^8b^9}\).
2. Rewrite the fourth root as the fourth root of each factor separately: \(^4\sqrt{256} \times ^4\sqrt{a^8} \times ^4\sqrt{b^9}\).
3. Simplify each part: - \(^4\sqrt{256}\) is the fourth root of 256, - \(^4\sqrt{a^8}\) can be simplified using the property \(^n\sqrt{x^m} = x^{m/n}\), - \(^4\sqrt{b^9}\) can be rewritten as \(^4\sqrt{b^8 \times b} = ^4\sqrt{b^8} \times ^4\sqrt{b}\).
4. Calculate the exponents after applying the root: - \(a^{8/4} = a^2\), - \(b^{8/4} = b^2\), - and keep \(^4\sqrt{b}\) as it is because \(b\) is left inside the root.
5. Combine all simplified parts: multiply the simplified fourth root of 256 by \(a^2\), \(b^2\), and the remaining \(^4\sqrt{b}\) to write the final simplified expression.