The expression root(3)(10^(4))*root(3)(10) is equivalent to 10^((4)/(9)) 100^((5)/(3)) 10^((5)/(3)) 100^((4)/(9))

Identify Equation and Apply Property: Identify the equation and apply the property of multiplying roots with the same index. The cube roots can be combined by multiplying the radicands (the numbers under the root). root(3)(10^(4))*root(3)(10) = root(3)(10^(4) * 10)Combine Cube Roots by Multiplying: Simplify the multiplication under the cube root. Multiply 10^(4) by 10. 10^(4) * 10 = 10^(4+1) = 10^(5)Simplify Multiplication: Write the simplified multiplication under the cube root. root(3)(10^(5))Write Simplified Multiplication: Convert the cube root to an exponent form. The cube root of a number is the same as raising that number to the power of 1/3. root(3)(10^(5)) = 10^(5/3)

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Q. The expression

\sqrt[3]{10^{4}} \cdot \sqrt[3]{10}

is equivalent to

\newline

10^{\frac{4}{9}}

\newline

100^{\frac{5}{3}}

\newline

10^{\frac{5}{3}}

\newline

100^{\frac{4}{9}}

Identify Equation and Apply Property: Identify the equation and apply the property of multiplying roots with the same index. $\newline$ The cube roots can be combined by multiplying the radicands (the numbers under the root). $\newline$ $\sqrt[3]{10^{4}}\cdot\sqrt[3]{10} = \sqrt[3]{10^{4} \cdot 10}$
Combine Cube Roots by Multiplying: Simplify the multiplication under the cube root. $\newline$ Multiply $10^{4}$ by $10$ . $\newline$ $10^{4} \times 10 = 10^{4+1} = 10^{5}$
Simplify Multiplication: Write the simplified multiplication under the cube root. $\sqrt[3]{10^{5}}$
Write Simplified Multiplication: Convert the cube root to an exponent form. $\newline$ The cube root of a number is the same as raising that number to the power of $1/3$ . $\newline$ $\sqrt[3]{10^{5}} = 10^{5/3}$