root(3)(x^(2))*x^(-(1)/(5)) Which of the following is equivalent to the given expression for all real values of x ? Choose 1 answer: (A) root(15)(x^(7)) (B) root(7)(x^(15)) (C) (1)/(sqrt(x^(15))) (D) (1)/(root(15)(x^(2)))

Given Expression: We are given the expression $\sqrt[3]{x^2} \cdot x^{-\frac{1}{5}}$. To simplify this expression, we need to combine the exponents by using the properties of exponents.Convert Cube Root to Exponent: First, we convert the cube root into an exponent. The cube root of $x^2$ can be written as $x^{\frac{2}{3}}$. So the expression becomes $x^{\frac{2}{3}} \cdot x^{-\frac{1}{5}}$.Add Exponents of Same Base: Next, we add the exponents of the same base $x$ when multiplying. The sum of the exponents is $\frac{2}{3} + (-\frac{1}{5})$. To add these fractions, we need a common denominator, which is 15. So we convert the fractions: $\frac{2}{3} = \frac{10}{15}$ and $-\frac{1}{5} = -\frac{3}{15}$.Convert Exponent Back to Radical Form: Now we add the converted exponents: $\frac{10}{15} + (-\frac{3}{15}) = \frac{10 - 3}{15} = \frac{7}{15}$. Therefore, the simplified expression is $x^{\frac{7}{15}}$.Match with Answer Choices: Finally, we convert the exponent back to a radical form. The expression $x^{\frac{7}{15}}$ is equivalent to the 15th root of $x^7$, which is written as $\sqrt[15]{x^7}$.Comparing the simplified expression with the answer choices, we find that it matches with choice (A) $\sqrt[15]{x^7}$.

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root(3)(x^(2))*x^(-(1)/(5))
Which of the following is equivalent to the given expression for all real values of
x ?
Choose 1 answer:
(A)
root(15)(x^(7))
(B)
root(7)(x^(15))
(C)
(1)/(sqrt(x^(15)))
(D)
(1)/(root(15)(x^(2)))

$\sqrt[3]{x^{2}} \cdot x^{-\frac{1}{5}}$ $\newline$ Which of the following is equivalent to the given expression for all real values of $x$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $\sqrt[15]{x^{7}}$ $\newline$ (B) $\sqrt[7]{x^{15}}$ $\newline$ (C) $\frac{1}{\sqrt{x^{15}}}$ $\newline$ (D) $\frac{1}{\sqrt[15]{x^{2}}}$

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Math Problems

Algebra 1

Compare linear and exponential growth

Full solution

\sqrt[3]{x^{2}} \cdot x^{-\frac{1}{5}}

\newline

Which of the following is equivalent to the given expression for all real values of

x

\newline

Choose

1

answer:

\newline

(A)

\sqrt[15]{x^{7}}

\newline

(B)

\sqrt[7]{x^{15}}

\newline

(C)

\frac{1}{\sqrt{x^{15}}}

\newline

(D)

\frac{1}{\sqrt[15]{x^{2}}}

Given Expression: We are given the expression $\sqrt[3]{x^2} \cdot x^{-\frac{1}{5}}$ . To simplify this expression, we need to combine the exponents by using the properties of exponents.
Convert Cube Root to Exponent: First, we convert the cube root into an exponent. The cube root of $x^2$ can be written as $x^{\frac{2}{3}}$ . So the expression becomes $x^{\frac{2}{3}} \cdot x^{-\frac{1}{5}}$ .
Add Exponents of Same Base: Next, we add the exponents of the same base $x$ when multiplying. The sum of the exponents is $\frac{2}{3} + (-\frac{1}{5})$ . To add these fractions, we need a common denominator, which is $15$ . So we convert the fractions: $\frac{2}{3} = \frac{10}{15}$ and $-\frac{1}{5} = -\frac{3}{15}$ .
Convert Exponent Back to Radical Form: Now we add the converted exponents: $\frac{10}{15} + (-\frac{3}{15}) = \frac{10 - 3}{15} = \frac{7}{15}$ . Therefore, the simplified expression is $x^{\frac{7}{15}}$ .
Match with Answer Choices: Finally, we convert the exponent back to a radical form. The expression $x^{\frac{7}{15}}$ is equivalent to the $15$ th root of $x^7$ , which is written as $\sqrt[15]{x^7}$ .
Match with Answer Choices: Finally, we convert the exponent back to a radical form. The expression $x^{\frac{7}{15}}$ is equivalent to the $15$ th root of $x^7$ , which is written as $\sqrt[15]{x^7}$ .Comparing the simplified expression with the answer choices, we find that it matches with choice (A) $\sqrt[15]{x^7}$ .