Solve x ^ (5/3) f . f(x) = 6x ^ (4/3) - 1/(3x ^ 3) - 1/2 * x ^ 2 | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$x ^ { \frac { 5 } { 3 } } f ( x ) = 6 x ^ { \frac { 4 } { 3 } } - \frac { 1 } { 3 x ^ { 3 } } - \frac { 1 } { 2 } x ^ { 2 }$$

Solve for f (complex solution)

$f=-\frac{x^{-\frac{2}{3}}}{2}+6x^{-\frac{4}{3}}-\frac{x^{-\frac{17}{3}}}{3}$
$x\neq 0$

Steps for Solving Linear Equation

Multiply both sides of the equation by $6x^{3}$, the least common multiple of $3x^{3},2$.

$$x^{\frac{5}{3}}fx\times 6x^{3}=6x^{\frac{4}{3}}\times 6x^{3}-2-\frac{1}{2}x^{2}\times 6x^{3}$$

To multiply powers of the same base, add their exponents. Add $\frac{5}{3}$ and $1$ to get $\frac{8}{3}$.

$$x^{\frac{8}{3}}f\times 6x^{3}=6x^{\frac{4}{3}}\times 6x^{3}-2-\frac{1}{2}x^{2}\times 6x^{3}$$

To multiply powers of the same base, add their exponents. Add $\frac{8}{3}$ and $3$ to get $\frac{17}{3}$.

$$x^{\frac{17}{3}}f\times 6=6x^{\frac{4}{3}}\times 6x^{3}-2-\frac{1}{2}x^{2}\times 6x^{3}$$

To multiply powers of the same base, add their exponents. Add $\frac{4}{3}$ and $3$ to get $\frac{13}{3}$.

$$x^{\frac{17}{3}}f\times 6=6x^{\frac{13}{3}}\times 6-2-\frac{1}{2}x^{2}\times 6x^{3}$$

To multiply powers of the same base, add their exponents. Add $2$ and $3$ to get $5$.

$$x^{\frac{17}{3}}f\times 6=6x^{\frac{13}{3}}\times 6-2-\frac{1}{2}x^{5}\times 6$$

Multiply $6$ and $6$ to get $36$.

$$x^{\frac{17}{3}}f\times 6=36x^{\frac{13}{3}}-2-\frac{1}{2}x^{5}\times 6$$

Multiply $-\frac{1}{2}$ and $6$ to get $-3$.

$$x^{\frac{17}{3}}f\times 6=36x^{\frac{13}{3}}-2-3x^{5}$$

The equation is in standard form.

$$6x^{\frac{17}{3}}f=-3x^{5}+36x^{\frac{13}{3}}-2$$

Divide both sides by $6x^{\frac{17}{3}}$.

$$\frac{6x^{\frac{17}{3}}f}{6x^{\frac{17}{3}}}=\frac{-3x^{5}+36x^{\frac{13}{3}}-2}{6x^{\frac{17}{3}}}$$

Dividing by $6x^{\frac{17}{3}}$ undoes the multiplication by $6x^{\frac{17}{3}}$.

$$f=\frac{-3x^{5}+36x^{\frac{13}{3}}-2}{6x^{\frac{17}{3}}}$$

Divide $36x^{\frac{13}{3}}-2-3x^{5}$ by $6x^{\frac{17}{3}}$.

$$f=-\frac{x^{-\frac{2}{3}}}{2}+6x^{-\frac{4}{3}}-\frac{x^{-\frac{17}{3}}}{3}$$

Solve for f

$f=-\frac{1}{2x^{\frac{2}{3}}}+\frac{6}{x^{\frac{4}{3}}}-\frac{1}{3x^{\frac{17}{3}}}$
$x\neq 0$

Steps for Solving Linear Equation

Multiply both sides of the equation by $6x^{3}$, the least common multiple of $3x^{3},2$.

$$x^{\frac{5}{3}}fx\times 6x^{3}=6x^{\frac{4}{3}}\times 6x^{3}-2-\frac{1}{2}x^{2}\times 6x^{3}$$

To multiply powers of the same base, add their exponents. Add $\frac{5}{3}$ and $1$ to get $\frac{8}{3}$.

$$x^{\frac{8}{3}}f\times 6x^{3}=6x^{\frac{4}{3}}\times 6x^{3}-2-\frac{1}{2}x^{2}\times 6x^{3}$$

To multiply powers of the same base, add their exponents. Add $\frac{8}{3}$ and $3$ to get $\frac{17}{3}$.

$$x^{\frac{17}{3}}f\times 6=6x^{\frac{4}{3}}\times 6x^{3}-2-\frac{1}{2}x^{2}\times 6x^{3}$$

To multiply powers of the same base, add their exponents. Add $\frac{4}{3}$ and $3$ to get $\frac{13}{3}$.

$$x^{\frac{17}{3}}f\times 6=6x^{\frac{13}{3}}\times 6-2-\frac{1}{2}x^{2}\times 6x^{3}$$

To multiply powers of the same base, add their exponents. Add $2$ and $3$ to get $5$.

$$x^{\frac{17}{3}}f\times 6=6x^{\frac{13}{3}}\times 6-2-\frac{1}{2}x^{5}\times 6$$

Multiply $6$ and $6$ to get $36$.

$$x^{\frac{17}{3}}f\times 6=36x^{\frac{13}{3}}-2-\frac{1}{2}x^{5}\times 6$$

Multiply $-\frac{1}{2}$ and $6$ to get $-3$.

$$x^{\frac{17}{3}}f\times 6=36x^{\frac{13}{3}}-2-3x^{5}$$

The equation is in standard form.

$$6x^{\frac{17}{3}}f=-3x^{5}+36x^{\frac{13}{3}}-2$$

Divide both sides by $6x^{\frac{17}{3}}$.

$$\frac{6x^{\frac{17}{3}}f}{6x^{\frac{17}{3}}}=\frac{-3x^{5}+36x^{\frac{13}{3}}-2}{6x^{\frac{17}{3}}}$$

Dividing by $6x^{\frac{17}{3}}$ undoes the multiplication by $6x^{\frac{17}{3}}$.

$$f=\frac{-3x^{5}+36x^{\frac{13}{3}}-2}{6x^{\frac{17}{3}}}$$

Divide $36x^{\frac{13}{3}}-2-3x^{5}$ by $6x^{\frac{17}{3}}$.

$$f=-\frac{1}{2x^{\frac{2}{3}}}+\frac{6}{x^{\frac{4}{3}}}-\frac{1}{3x^{\frac{17}{3}}}$$