Which of the following expressions is equivalent to the expression above?sqrt[4]{a^4b^8}

1. TRANSLATE the radical to exponential form

• Given: \(\sqrt[4]{\mathrm{a}^4\mathrm{b}^8}\)
• Convert to fractional exponent: \((\mathrm{a}^4\mathrm{b}^8)^{1/4}\)
• This makes applying exponent rules much easier

2. SIMPLIFY by distributing the fractional exponent

• Apply the property: \((\mathrm{xy})^\mathrm{n} = \mathrm{x}^\mathrm{n} \cdot \mathrm{y}^\mathrm{n}\)
• \((\mathrm{a}^4\mathrm{b}^8)^{1/4} = (\mathrm{a}^4)^{1/4} \cdot (\mathrm{b}^8)^{1/4}\)
• Now we can work with each variable separately

3. SIMPLIFY each term using the power rule

• For the first term: \((\mathrm{a}^4)^{1/4} = \mathrm{a}^{4 \times 1/4} = \mathrm{a}^1 = \mathrm{a}\)
• For the second term: \((\mathrm{b}^8)^{1/4} = \mathrm{b}^{8 \times 1/4} = \mathrm{b}^2\)
• Remember: when raising a power to a power, multiply the exponents

4. Combine the simplified terms

• \(\mathrm{a} \cdot \mathrm{b}^2 = \mathrm{ab}^2\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make arithmetic errors when multiplying fractions and exponents, particularly confusing \(8 \times 1/4 = 2\) with other calculations like \(8 \div 4 = 2\) or getting \(8 \times 1/4 = 8/4 = 2\) but writing it as \(\mathrm{b}^4\) instead of \(\mathrm{b}^2\).

This may lead them to select Choice (D) \((\mathrm{ab}^3)\) or get confused about the final exponent values.

Second Most Common Error:

Poor TRANSLATE reasoning: Students try to work directly with the radical without converting to exponential form, leading to confusion about how to systematically apply exponent rules to the expression under the radical.

This leads to confusion and guessing among the answer choices.

The Bottom Line:

Converting radicals to exponential form transforms a potentially confusing problem into straightforward exponent rule application. The key insight is recognizing that \(\sqrt[4]{\mathrm{x}} = \mathrm{x}^{1/4}\), which then allows systematic use of the power rule.