Apply definition of fourth root: Apply the definition of the fourth root to root(4)(16ab^(4)). The fourth root of a product is the product of the fourth roots of each factor.Find fourth root of 16: Find the fourth root of 16. The fourth root of 16, which is 2^4, is 2 because (2^4)^(1/4) = 2^(4*(1/4)) = 2^1 = 2.Find fourth root of ab^4: Find the fourth root of ab^4. The fourth root of ab^4 is a^(1/4)b because b^4 raised to the power of 1/4 cancels out, leaving a^(1/4)b^1 = a^(1/4)b.Combine results: Combine the results from the previous steps. The fourth root of 16ab^4 is the product of the fourth root of 16 and the fourth root of ab^4, which is 2 * a^(1/4)b.

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$\sqrt[4]{16 a b^{4}}$

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Algebra 2

Roots of rational numbers

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\sqrt[4]{16 a b^{4}}

Apply definition of fourth root: Apply the definition of the fourth root to $\sqrt[4]{16a b^{4}}$ . The fourth root of a product is the product of the fourth roots of each factor.
Find fourth root of $16$ : Find the fourth root of $16$ . $\newline$ The fourth root of $16$ , which is $2^4$ , is $2$ because $(2^4)^{1/4} = 2^{4*(1/4)} = 2^1 = 2$ .
Find fourth root of $ab^4$ : Find the fourth root of $ab^4$ . The fourth root of $ab^4$ is $a^{\frac{1}{4}}b$ because $b^4$ raised to the power of $\frac{1}{4}$ cancels out, leaving $a^{\frac{1}{4}}b^1 = a^{\frac{1}{4}}b$ .
Combine results: Combine the results from the previous steps. $\newline$ The fourth root of $16ab^4$ is the product of the fourth root of $16$ and the fourth root of $ab^4$ , which is $2 \times a^{1/4}b$ .