Select the equivalent expression. root(33)((1)/(k^(8)*k^(3))) (A) k^(-3) (B) k^(3) (C) k^((1)/(3)) (D) k^(-(1)/(3))

Understand given expression: Understand the given expression. The given expression is root(33)((1)/(k^(8)*k^(3))). This means we are looking for the 33rd root of the fraction (1)/(k^(8)*k^(3)).Combine exponents in denominator: Combine the exponents in the denominator. When multiplying powers with the same base, we add the exponents. So, k^(8)*k^(3) = k^(8+3) = k^(11). Now the expression becomes root(33)((1)/(k^(11))).Convert radical expression to exponent form: Convert the radical expression to an exponent form. The 33rd root of a number can be written as that number raised to the power of 1/33. So, root(33)((1)/(k^(11))) becomes (1/k^(11))^(1/33).Apply power of power rule: Apply the power of a power rule. When raising a power to a power, we multiply the exponents. So, (1/k^(11))^(1/33) becomes 1/k^(11*(1/33)).Simplify exponent: Simplify the exponent. Multiply the exponents: 11*(1/33) = 11/33. Simplify the fraction: 11/33 = 1/3. Now the expression is 1/k^(1/3).Write expression with negative exponent: Write the expression with a negative exponent. The expression 1/k^(1/3) can be written as k^(-1/3).Match final expression with options: Match the final expression with the given options. The final expression we found is k^(-1/3), which matches one of the given options.

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Select the equivalent expression.
root(33)((1)/(k^(8)*k^(3)))
(A) k^(-3)
(B) k^(3)
(C) k^((1)/(3))
(D) k^(-(1)/(3))

Select the equivalent expression. $\newline$ $\sqrt[33]{\frac{1}{k^{8} \cdot k^{3}}}$ $\newline$ (A) $k^{-3}$ $\newline$ (B) $k^{3}$ r $\newline$ (C) $k^{\frac{1}{3}}$ $\newline$ (D) $k^{-\frac{1}{3}}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. Select the equivalent expression.

\newline

\sqrt[33]{\frac{1}{k^{8} \cdot k^{3}}}

\newline

(A)

k^{-3}

\newline

(B)

k^{3}

\newline

(C)

k^{\frac{1}{3}}

\newline

(D)

k^{-\frac{1}{3}}

Understand given expression: Understand the given expression. $\newline$ The given expression is $\sqrt[33]{\frac{1}{k^{8}*k^{3}}}$ . $\newline$ This means we are looking for the $33^{\text{rd}}$ root of the fraction $\frac{1}{k^{8}*k^{3}}$ .
Combine exponents in denominator: Combine the exponents in the denominator. $\newline$ When multiplying powers with the same base, we add the exponents. $\newline$ So, $k^{8}\times k^{3} = k^{8+3} = k^{11}$ . $\newline$ Now the expression becomes $\sqrt[33]{\frac{1}{k^{11}}}$ .
Convert radical expression to exponent form: Convert the radical expression to an exponent form. $\newline$ The $33^{\text{rd}}$ root of a number can be written as that number raised to the power of $1/33$ . $\newline$ So, $\sqrt[33]{\frac{1}{k^{11}}}$ becomes $\left(\frac{1}{k^{11}}\right)^{1/33}$ .
Apply power of power rule: Apply the power of a power rule. $\newline$ When raising a power to a power, we multiply the exponents. $\newline$ So, $(1/k^{11})^{1/33}$ becomes $1/k^{11*(1/33)}$ .
Simplify exponent: Simplify the exponent. $\newline$ Multiply the exponents: $11*(1/33) = 11/33$ . $\newline$ Simplify the fraction: $11/33 = 1/3$ . $\newline$ Now the expression is $1/k^{1/3}$ .
Write expression with negative exponent: Write the expression with a negative exponent. $\newline$ The expression $1/k^{1/3}$ can be written as $k^{-1/3}$ .
Match final expression with options: Match the final expression with the given options. $\newline$ The final expression we found is $k^{(-1/3)}$ , which matches one of the given options.