The expression ((m^(2))/(m^((1)/(3))))^(-(1)/(2)) is equivalent to (1) -root(1)(m^(5)) (2) (1)/(root(6)(m^(5))) (3) -mroot(5)(m) (4) (1)/(mroot(3)(m))

Simplify Inside Parentheses: Simplify the expression inside the parentheses. We have the expression ((m^(2))/(m^((1)/(3))))^(-(1)/(2)). To simplify the expression inside the parentheses, we use the property of exponents that states when dividing like bases, we subtract the exponents. So, (m^(2))/(m^((1)/(3))) = m^(2 - 1/3) = m^(6/3 - 1/3) = m^(5/3).Apply Negative Exponent: Apply the negative exponent outside the parentheses. Now we have (m^(5/3))^(-(1)/(2)). A negative exponent means we take the reciprocal of the base. Therefore, we have (m^(5/3))^(-(1)/(2)) = (1/(m^(5/3)))^(1/2).Simplify Fractional Exponent: Simplify the expression with the fractional exponent. We have (1/(m^(5/3)))^(1/2). When we raise a power to a power, we multiply the exponents. So, we get 1/(m^(5/3 * 1/2)) = 1/(m^(5/6)).Use Radical Notation: Write the expression using radical notation. The expression 1/(m^(5/6)) can be written in radical notation as (1)/(root(6)(m^(5))).Match with Options: Match the expression to the given options. The expression (1)/(root(6)(m^(5))) matches option (2).

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The expression $\left(\frac{m^{2}}{m^{\left(\frac{1}{3}\right)}}\right)^{-\left(\frac{1}{2}\right)}$ is equivalent to $\newline$ ( $1$ ) $-\sqrt[1]{m^{5}}$ $\newline$ ( $2$ ) $\frac{1}{\sqrt[6]{m^{5}}}$ $\newline$ ( $3$ ) $-m\sqrt[5]{m}$ $\newline$ ( $4$ ) $\frac{1}{\sqrt[3]{m}}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. The expression

\left(\frac{m^{2}}{m^{\left(\frac{1}{3}\right)}}\right)^{-\left(\frac{1}{2}\right)}

is equivalent to

\newline

(

1

)

-\sqrt[1]{m^{5}}

\newline

(

2

)

\frac{1}{\sqrt[6]{m^{5}}}

\newline

(

3

)

-m\sqrt[5]{m}

\newline

(

4

)

\frac{1}{\sqrt[3]{m}}

Simplify Inside Parentheses: Simplify the expression inside the parentheses. $\newline$ We have the expression $((m^{2})/(m^{(1)/(3)}))^{-(1)/(2)}$ . To simplify the expression inside the parentheses, we use the property of exponents that states when dividing like bases, we subtract the exponents. $\newline$ So, $(m^{2})/(m^{(1)/(3)}) = m^{2 - 1/3} = m^{6/3 - 1/3} = m^{5/3}.$
Apply Negative Exponent: Apply the negative exponent outside the parentheses. $\newline$ Now we have $(m^{5/3})^{-(1)/(2)}$ . A negative exponent means we take the reciprocal of the base. Therefore, we have $(m^{5/3})^{-(1)/(2)} = (1/(m^{5/3}))^{1/2}$ .
Simplify Fractional Exponent: Simplify the expression with the fractional exponent. $\newline$ We have $(1/(m^{5/3}))^{1/2}$ . When we raise a power to a power, we multiply the exponents. So, we get $1/(m^{5/3 \cdot 1/2}) = 1/(m^{5/6})$ .
Use Radical Notation: Write the expression using radical notation. $\newline$ The expression $\frac{1}{m^{\frac{5}{6}}}$ can be written in radical notation as $\frac{1}{\sqrt[6]{m^{5}}}$ .
Match with Options: Match the expression to the given options. $\newline$ The expression $(1)/(\sqrt[6]{m^{5}})$ matches option $(2)$ .