Find All Complex Solutions 4x^5-100x^(13/3)=0

$4 x^{5} - 100 x^{\frac{13}{3}} = 0$

Step 1

Factor the left side of the equation.

Factor $4 x^{\frac{13}{3}}$ out of $4 x^{5} - 100 x^{\frac{13}{3}}$ .

Factor $4 x^{\frac{13}{3}}$ out of $4 x^{5}$ .

$4 x^{\frac{13}{3}} (x^{\frac{2}{3}}) - 100 x^{\frac{13}{3}} = 0$

Factor $4 x^{\frac{13}{3}}$ out of $- 100 x^{\frac{13}{3}}$ .

$4 x^{\frac{13}{3}} (x^{\frac{2}{3}}) + 4 x^{\frac{13}{3}} (- 25) = 0$

Factor $4 x^{\frac{13}{3}}$ out of $4 x^{\frac{13}{3}} (x^{\frac{2}{3}}) + 4 x^{\frac{13}{3}} (- 25)$ .

$4 x^{\frac{13}{3}} (x^{\frac{2}{3}} - 25) = 0$

Rewrite $x^{\frac{2}{3}}$ as ${(x^{\frac{1}{3}})}^{2}$ .

$4 x^{\frac{13}{3}} ({(x^{\frac{1}{3}})}^{2} - 25) = 0$

Rewrite $25$ as $5^{2}$ .

$4 x^{\frac{13}{3}} ({(x^{\frac{1}{3}})}^{2} - 5^{2}) = 0$

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{\frac{1}{3}}$ and $b = 5$ .

$4 x^{\frac{13}{3}} ((x^{\frac{1}{3}} + 5) (x^{\frac{1}{3}} - 5)) = 0$

Remove unnecessary parentheses.

$4 x^{\frac{13}{3}} (x^{\frac{1}{3}} + 5) (x^{\frac{1}{3}} - 5) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{\frac{13}{3}} = 0$

$x^{\frac{1}{3}} + 5 = 0$

$x^{\frac{1}{3}} - 5 = 0$

Step 3

Set $x^{\frac{13}{3}}$ equal to $0$ and solve for $x$ .

Set $x^{\frac{13}{3}}$ equal to $0$ .

$x^{\frac{13}{3}} = 0$

Solve $x^{\frac{13}{3}} = 0$ for $x$ .

Raise each side of the equation to the power of $\frac{3}{13}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{13}{3}})}^{\frac{3}{13}} = 0^{\frac{3}{13}}$

Simplify the exponent.

Simplify the left side.

Simplify ${(x^{\frac{13}{3}})}^{\frac{3}{13}}$ .

Multiply the exponents in ${(x^{\frac{13}{3}})}^{\frac{3}{13}}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{13}{3} \cdot \frac{3}{13}} = 0^{\frac{3}{13}}$

Cancel the common factor of $13$ .

Cancel the common factor.

$x^{\frac{13}{3} \cdot \frac{3}{13}} = 0^{\frac{3}{13}}$

Rewrite the expression.

$x^{\frac{1}{3} \cdot 3} = 0^{\frac{3}{13}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = 0^{\frac{3}{13}}$

Rewrite the expression.

$x^{1} = 0^{\frac{3}{13}}$

Simplify.

$x = 0^{\frac{3}{13}}$

Simplify the right side.

Simplify $0^{\frac{3}{13}}$ .

Simplify the expression.

Rewrite $0$ as $0^{13}$ .

$x = {(0^{13})}^{\frac{3}{13}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = 0^{13 (\frac{3}{13})}$

Cancel the common factor of $13$ .

Cancel the common factor.

$x = 0^{13 (\frac{3}{13})}$

Rewrite the expression.

$x = 0^{3}$

Raising $0$ to any positive power yields $0$ .

$x = 0$

Step 4

Set $x^{\frac{1}{3}} + 5$ equal to $0$ and solve for $x$ .

Set $x^{\frac{1}{3}} + 5$ equal to $0$ .

$x^{\frac{1}{3}} + 5 = 0$

Solve $x^{\frac{1}{3}} + 5 = 0$ for $x$ .

Subtract $5$ from both sides of the equation.

$x^{\frac{1}{3}} = - 5$

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{3}})}^{3} = {(- 5)}^{3}$

Simplify the exponent.

Simplify the left side.

Simplify ${(x^{\frac{1}{3}})}^{3}$ .

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{3} \cdot 3} = {(- 5)}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = {(- 5)}^{3}$

Rewrite the expression.

$x^{1} = {(- 5)}^{3}$

Simplify.

$x = {(- 5)}^{3}$

Simplify the right side.

Raise $- 5$ to the power of $3$ .

$x = - 125$

Step 5

Set $x^{\frac{1}{3}} - 5$ equal to $0$ and solve for $x$ .

Set $x^{\frac{1}{3}} - 5$ equal to $0$ .

$x^{\frac{1}{3}} - 5 = 0$

Solve $x^{\frac{1}{3}} - 5 = 0$ for $x$ .

Add $5$ to both sides of the equation.

$x^{\frac{1}{3}} = 5$

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{3}})}^{3} = 5^{3}$

Simplify the exponent.

Simplify the left side.

Simplify ${(x^{\frac{1}{3}})}^{3}$ .

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{3} \cdot 3} = 5^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = 5^{3}$

Rewrite the expression.

$x^{1} = 5^{3}$

Simplify.

$x = 5^{3}$

Simplify the right side.

Raise $5$ to the power of $3$ .

$x = 125$

Step 6

The final solution is all the values that make $4 x^{\frac{13}{3}} (x^{\frac{1}{3}} + 5) (x^{\frac{1}{3}} - 5) = 0$ true.

$x = 0, - 125, 125$

Find All Complex Solutions 4x^5-100x^(13/3)=0

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