(16a^(6)b^(2))^((1)/(2))

Identify base and exponents: Identify the base and the exponents in the expression (16a^(6)b^(2))^((1)/(2)). In this expression, the base is 16a^(6)b^(2) and the exponent is (1)/(2), which indicates we are looking for the square root of the base.Apply power of power rule: Apply the power of a power rule, which states that (x^m)^n = x^(m*n), to each part of the base. (16a^(6)b^(2))^((1)/(2)) = 16^((1)/(2)) * a^(6*(1)/(2)) * b^(2*(1)/(2))Simplify each part: Simplify each part separately. 16^((1)/(2)) is the square root of 16, which is 4. a^(6*(1)/(2)) simplifies to a^(3) because 6*(1)/(2) = 3. b^(2*(1)/(2)) simplifies to b^(1) or simply b because 2*(1)/(2) = 1.Combine simplified parts: Combine the simplified parts to get the final answer. The simplified form of the expression is 4a^(3)b.

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$\left(16 a^{6} b^{2}\right)^{\frac{1}{2}}$

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Algebra 2

Evaluate rational exponents

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\left(16 a^{6} b^{2}\right)^{\frac{1}{2}}

Identify base and exponents: Identify the base and the exponents in the expression $(16a^{6}b^{2})^{\frac{1}{2}}$ . In this expression, the base is $16a^{6}b^{2}$ and the exponent is $\frac{1}{2}$ , which indicates we are looking for the square root of the base.
Apply power of power rule: Apply the power of a power rule, which states that $(x^m)^n = x^{m*n}$ , to each part of the base. $\newline$ $(16a^{6}b^{2})^{\frac{1}{2}} = 16^{\frac{1}{2}} * a^{6*\frac{1}{2}} * b^{2*\frac{1}{2}}$
Simplify each part: Simplify each part separately. $\newline$ $16^{\frac{1}{2}}$ is the square root of $16$ , which is $4$ . $\newline$ $a^{6\cdot\frac{1}{2}}$ simplifies to $a^{3}$ because $6\cdot\frac{1}{2} = 3$ . $\newline$ $b^{2\cdot\frac{1}{2}}$ simplifies to $b^{1}$ or simply $b$ because $2\cdot\frac{1}{2} = 1$ .
Combine simplified parts: Combine the simplified parts to get the final answer. $\newline$ The simplified form of the expression is $4a^{3}b$ .