Your answer is incorrect. The expression can be simplified further. $\newline$ Write the following expression in simplified radical form. $\newline$ $\sqrt[4]{80 s^{10} t^{16}}$ $\newline$ Assume that all of the variables in the expression represent positive real numbers. $\newline$ $2 s^{2} r^{3} \sqrt{5 s^{2} t^{4}}$ $\newline$ $\sqrt{\square} \quad \sqrt[\square]{\square} \quad \square$

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Algebra 1

Power rule with rational exponents

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Q. Your answer is incorrect. The expression can be simplified further.

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Write the following expression in simplified radical form.

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\sqrt[4]{80 s^{10} t^{16}}

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Assume that all of the variables in the expression represent positive real numbers.

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2 s^{2} r^{3} \sqrt{5 s^{2} t^{4}}

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\sqrt{\square} \quad \sqrt[\square]{\square} \quad \square

Identify Properties: We have the expression: $\sqrt[4]{80s^{10}t^{16}}$ Which properties of radicals and exponents can be used to simplify the expression? To simplify this expression, we can use the property that the fourth root of a product is the product of the fourth roots. Also, we can simplify the expression by breaking down the number $80$ into its prime factors and separating the variables with even exponents that are multiples of $4$ .
Factorize $80$ : First, let's factor $80$ into its prime factors. $\newline$ $80 = 2 \times 40 = 2 \times 2 \times 20 = 2 \times 2 \times 2 \times 10 = 2 \times 2 \times 2 \times 2 \times 5 = 2^4 \times 5$ $\newline$ Now we can rewrite the expression using these factors. $\newline$ $\sqrt[4]{80s^{10}t^{16}} = \sqrt[4]{2^4 \times 5 \times s^{10} \times t^{16}}$
Apply Fourth Root: Next, we apply the fourth root to each factor separately. $\newline$ The fourth root of $2^4$ is $2$ , because $(2^4)^{1/4} = 2^{4*(1/4)} = 2^1 = 2$ . $\newline$ The fourth root of $s^{10}$ is $s^{10/4} = s^{5/2}$ , because $(s^{10})^{1/4} = s^{10*(1/4)} = s^{5/2}$ . $\newline$ The fourth root of $t^{16}$ is $t^{16/4} = t^4$ , because $(t^{16})^{1/4} = t^{16*(1/4)} = t^4$ . $\newline$ The fourth root of $5$ remains under the radical because it cannot be simplified further. $\newline$ $2$ $0$
Express in Radical Form: Now, let's express $s^{\frac{5}{2}}$ in radical form. $\newline$ $s^{\frac{5}{2}} = s^{2 + \frac{1}{2}} = s^2 \cdot s^{\frac{1}{2}} = s^2 \cdot \sqrt{s}$ $\newline$ So we can rewrite the expression as: $\newline$ $2 \cdot s^2 \cdot \sqrt{s} \cdot t^4 \cdot \sqrt[4]{5}$
Combine Terms: Finally, we combine all the terms together to get the simplified expression. $\newline$ The simplified expression is: $\newline$ $2s^2 \cdot \sqrt{s} \cdot t^4 \cdot \sqrt[4]{5}$