f(x)=(5)/(root(3)(2x-x^(2)))

Understand and Identify Cube Root: We are given the function f(x)=(5)/(root(3)(2x-x^(2))). The first step is to understand the function and identify the cube root in the denominator.Rewrite Cube Root as Fractional Exponent: The cube root of a variable expression can be written as a fractional exponent. So, we rewrite the cube root in the denominator as a power of 1/3. f(x) = 5 / (2x - x^2)^(1/3)Check and Simplify Quadratic Expression: Now, we need to check if the expression inside the cube root, 2x - x^2, can be simplified further. This is a quadratic expression, and we should look for common factors or if it can be factored.Factorize Quadratic Expression: The expression 2x - x^2 can be factored by taking x as a common factor. x(2 - x) = 2x - x^2Substitute Factored Form: Now we substitute the factored form back into the function. f(x) = 5 / (x(2 - x))^(1/3)Apply Property of Exponents: Since we have factored the expression inside the cube root, we can now apply the property of exponents which states that (ab)^(1/3) = a^(1/3) * b^(1/3). f(x) = 5 / (x^(1/3) * (2 - x)^(1/3))Final Simplification: The function is now simplified as much as possible. We cannot simplify it further without knowing the specific value of x.

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$f(x)=\frac{5}{\sqrt[3]{2x-x^{2}}}$

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Algebra 1

Multiplication with rational exponents

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f(x)=\frac{5}{\sqrt[3]{2x-x^{2}}}

Understand and Identify Cube Root: We are given the function $f(x)=\frac{5}{\sqrt[3]{2x-x^{2}}}$ . The first step is to understand the function and identify the cube root in the denominator.
Rewrite Cube Root as Fractional Exponent: The cube root of a variable expression can be written as a fractional exponent. So, we rewrite the cube root in the denominator as a power of $\frac{1}{3}$ . $\newline$ $f(x) = \frac{5}{(2x - x^2)^{\frac{1}{3}}}$
Check and Simplify Quadratic Expression: Now, we need to check if the expression inside the cube root, $2x - x^2$ , can be simplified further. This is a quadratic expression, and we should look for common factors or if it can be factored.
Factorize Quadratic Expression: The expression $2x - x^2$ can be factored by taking $x$ as a common factor. $\newline$ $x(2 - x) = 2x - x^2$
Substitute Factored Form: Now we substitute the factored form back into the function. $\newline$ $f(x) = \frac{5}{(x(2 - x))^{\frac{1}{3}}}$
Apply Property of Exponents: Since we have factored the expression inside the cube root, we can now apply the property of exponents which states that $(ab)^{\frac{1}{3}} = a^{\frac{1}{3}} \times b^{\frac{1}{3}}$ . $f(x) = \frac{5}{x^{\frac{1}{3}} \times (2 - x)^{\frac{1}{3}}}$
Final Simplification: The function is now simplified as much as possible. We cannot simplify it further without knowing the specific value of $x$ .