Find the Roots (Zeros) 4x^5-100x^(13/3)=0

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

Step 1

Find a common factor $x^{\frac{13}{3}}$ that is present in each term.

$4 {(x^{\frac{13}{3}})}^{\frac{15}{13}} - 100 x^{\frac{13}{3}}$

Step 2

Substitute $u$ for $x^{\frac{13}{3}}$ .

$4 {(u)}^{\frac{15}{13}} - 100 u = 0$

Step 3

Solve for $u$ .

Factor the left side of the equation.

Factor $4 u$ out of $4 u^{\frac{15}{13}} - 100 u$ .

Factor $4 u$ out of $4 u^{\frac{15}{13}}$ .

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

Factor $4 u$ out of $- 100 u$ .

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

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

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

Rewrite $u^{\frac{2}{13}}$ as ${(u^{\frac{1}{13}})}^{2}$ .

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

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

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

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = u^{\frac{1}{13}}$ and $b = 5$ .

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

Remove unnecessary parentheses.

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

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

$u = 0$

$u^{\frac{1}{13}} + 5 = 0$

$u^{\frac{1}{13}} - 5 = 0$

Set $u$ equal to $0$ .

$u = 0$

Set $u^{\frac{1}{13}} + 5$ equal to $0$ and solve for $u$ .

Set $u^{\frac{1}{13}} + 5$ equal to $0$ .

$u^{\frac{1}{13}} + 5 = 0$

Solve $u^{\frac{1}{13}} + 5 = 0$ for $u$ .

Subtract $5$ from both sides of the equation.

$u^{\frac{1}{13}} = - 5$

Raise each side of the equation to the power of $13$ to eliminate the fractional exponent on the left side.

${(u^{\frac{1}{13}})}^{13} = {(- 5)}^{13}$

Simplify the exponent.

Simplify the left side.

Simplify ${(u^{\frac{1}{13}})}^{13}$ .

Multiply the exponents in ${(u^{\frac{1}{13}})}^{13}$ .

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

$u^{\frac{1}{13} \cdot 13} = {(- 5)}^{13}$

Cancel the common factor of $13$ .

Cancel the common factor.

$u^{\frac{1}{13} \cdot 13} = {(- 5)}^{13}$

Rewrite the expression.

$u^{1} = {(- 5)}^{13}$

Simplify.

$u = {(- 5)}^{13}$

Simplify the right side.

Raise $- 5$ to the power of $13$ .

$u = - 1220703125$

Set $u^{\frac{1}{13}} - 5$ equal to $0$ and solve for $u$ .

Set $u^{\frac{1}{13}} - 5$ equal to $0$ .

$u^{\frac{1}{13}} - 5 = 0$

Solve $u^{\frac{1}{13}} - 5 = 0$ for $u$ .

Add $5$ to both sides of the equation.

$u^{\frac{1}{13}} = 5$

Raise each side of the equation to the power of $13$ to eliminate the fractional exponent on the left side.

${(u^{\frac{1}{13}})}^{13} = 5^{13}$

Simplify the exponent.

Simplify the left side.

Simplify ${(u^{\frac{1}{13}})}^{13}$ .

Multiply the exponents in ${(u^{\frac{1}{13}})}^{13}$ .

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

$u^{\frac{1}{13} \cdot 13} = 5^{13}$

Cancel the common factor of $13$ .

Cancel the common factor.

$u^{\frac{1}{13} \cdot 13} = 5^{13}$

Rewrite the expression.

$u^{1} = 5^{13}$

Simplify.

$u = 5^{13}$

Simplify the right side.

Raise $5$ to the power of $13$ .

$u = 1220703125$

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

$u = 0, - 1220703125, 1220703125$

Step 4

Substitute $x$ for $u$ .

$x^{\frac{13}{3}} = 0, - 1220703125, 1220703125$

Step 5

Solve for $x^{\frac{13}{3}} = 0$ 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 6

Solve for $x^{\frac{13}{3}} = - 1220703125$ for $x$ .

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

${(x^{\frac{13}{3}})}^{\frac{3}{13}} = {(- 1220703125)}^{\frac{3}{13}}$

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, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{13}{3} \cdot \frac{3}{13}} = {(- 1220703125)}^{\frac{3}{13}}$

Cancel the common factor of $13$ .

Cancel the common factor.

$x^{\frac{13}{3} \cdot \frac{3}{13}} = {(- 1220703125)}^{\frac{3}{13}}$

Rewrite the expression.

$x^{\frac{1}{3} \cdot 3} = {(- 1220703125)}^{\frac{3}{13}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = {(- 1220703125)}^{\frac{3}{13}}$

Rewrite the expression.

$x^{1} = {(- 1220703125)}^{\frac{3}{13}}$

Simplify.

$x = {(- 1220703125)}^{\frac{3}{13}}$

Simplify the right side.

Simplify ${(- 1220703125)}^{\frac{3}{13}}$ .

Simplify the expression.

Rewrite $- 1220703125$ as ${(- 5)}^{13}$ .

$x = {({(- 5)}^{13})}^{\frac{3}{13}}$

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

$x = {(- 5)}^{13 (\frac{3}{13})}$

Cancel the common factor of $13$ .

Cancel the common factor.

$x = {(- 5)}^{13 (\frac{3}{13})}$

Rewrite the expression.

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

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

$x = - 125$

Step 7

Solve for $x^{\frac{13}{3}} = 1220703125$ for $x$ .

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

${(x^{\frac{13}{3}})}^{\frac{3}{13}} = 1220703125^{\frac{3}{13}}$

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, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{13}{3} \cdot \frac{3}{13}} = 1220703125^{\frac{3}{13}}$

Cancel the common factor of $13$ .

Cancel the common factor.

$x^{\frac{13}{3} \cdot \frac{3}{13}} = 1220703125^{\frac{3}{13}}$

Rewrite the expression.

$x^{\frac{1}{3} \cdot 3} = 1220703125^{\frac{3}{13}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = 1220703125^{\frac{3}{13}}$

Rewrite the expression.

$x^{1} = 1220703125^{\frac{3}{13}}$

Simplify.

$x = 1220703125^{\frac{3}{13}}$

Simplify the right side.

Simplify $1220703125^{\frac{3}{13}}$ .

Simplify the expression.

Rewrite $1220703125$ as $5^{13}$ .

$x = {(5^{13})}^{\frac{3}{13}}$

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

$x = 5^{13 (\frac{3}{13})}$

Cancel the common factor of $13$ .

Cancel the common factor.

$x = 5^{13 (\frac{3}{13})}$

Rewrite the expression.

$x = 5^{3}$

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

$x = 125$

Step 8

List all of the solutions.

$x = 0, - 125, 125$

Step 9

Exclude the solutions that do not make $4 x^{5} - 100 x^{\frac{13}{3}} = 0$ true.

$x = 0$

Step 10

Find the Roots (Zeros) 4x^5-100x^(13/3)=0

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