Algebra Examples

$2 x^{\frac{16}{5}} - 16 x^{\frac{13}{5}} = 0$

Step 1

Find a common factor $x^{\frac{13}{5}}$ that is present in each term.

$2 {(x^{\frac{13}{5}})}^{\frac{16}{13}} - 16 x^{\frac{13}{5}}$

Step 2

Substitute $u$ for $x^{\frac{13}{5}}$ .

$2 {(u)}^{\frac{16}{13}} - 16 u = 0$

Step 3

Solve for $u$ .

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Step 3.1

Factor the left side of the equation.

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Step 3.1.1

Factor $2 u$ out of $2 u^{\frac{16}{13}} - 16 u$ .

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Step 3.1.1.1

Factor $2 u$ out of $2 u^{\frac{16}{13}}$ .

$2 u (u^{\frac{3}{13}}) - 16 u = 0$

Step 3.1.1.2

Factor $2 u$ out of $- 16 u$ .

$2 u (u^{\frac{3}{13}}) + 2 u (- 8) = 0$

Step 3.1.1.3

Factor $2 u$ out of $2 u (u^{\frac{3}{13}}) + 2 u (- 8)$ .

$2 u (u^{\frac{3}{13}} - 8) = 0$

Step 3.1.2

Rewrite $u^{\frac{3}{13}}$ as ${(u^{\frac{1}{13}})}^{3}$ .

$2 u ({(u^{\frac{1}{13}})}^{3} - 8) = 0$

Step 3.1.3

Rewrite $8$ as $2^{3}$ .

$2 u ({(u^{\frac{1}{13}})}^{3} - 2^{3}) = 0$

Step 3.1.4

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = u^{\frac{1}{13}}$ and $b = 2$ .

$2 u ((u^{\frac{1}{13}} - 2) ({(u^{\frac{1}{13}})}^{2} + u^{\frac{1}{13}} \cdot 2 + 2^{2})) = 0$

Step 3.1.5

Factor.

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Step 3.1.5.1

Simplify.

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Step 3.1.5.1.1

Multiply the exponents in ${(u^{\frac{1}{13}})}^{2}$ .

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Step 3.1.5.1.1.1

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

$2 u ((u^{\frac{1}{13}} - 2) (u^{\frac{1}{13} \cdot 2} + u^{\frac{1}{13}} \cdot 2 + 2^{2})) = 0$

Step 3.1.5.1.1.2

Combine $\frac{1}{13}$ and $2$ .

$2 u ((u^{\frac{1}{13}} - 2) (u^{\frac{2}{13}} + u^{\frac{1}{13}} \cdot 2 + 2^{2})) = 0$

Step 3.1.5.1.2

Move $2$ to the left of $u^{\frac{1}{13}}$ .

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

Step 3.1.5.1.3

Raise $2$ to the power of $2$ .

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

Step 3.1.5.1.4

Reorder terms.

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

Step 3.1.5.2

Remove unnecessary parentheses.

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

Step 3.2

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

$u = 0$

$u^{\frac{1}{13}} - 2 = 0$

$2 u^{\frac{1}{13}} + u^{\frac{2}{13}} + 4 = 0$

Step 3.3

Set $u$ equal to $0$ .

$u = 0$

Step 3.4

Set $u^{\frac{1}{13}} - 2$ equal to $0$ and solve for $u$ .

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Step 3.4.1

Set $u^{\frac{1}{13}} - 2$ equal to $0$ .

$u^{\frac{1}{13}} - 2 = 0$

Step 3.4.2

Solve $u^{\frac{1}{13}} - 2 = 0$ for $u$ .

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Step 3.4.2.1

Add $2$ to both sides of the equation.

$u^{\frac{1}{13}} = 2$

Step 3.4.2.2

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

${(u^{\frac{1}{13}})}^{13} = 2^{13}$

Step 3.4.2.3

Simplify the exponent.

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Step 3.4.2.3.1

Simplify the left side.

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Step 3.4.2.3.1.1

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

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Step 3.4.2.3.1.1.1

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

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Step 3.4.2.3.1.1.1.1

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

$u^{\frac{1}{13} \cdot 13} = 2^{13}$

Step 3.4.2.3.1.1.1.2

Cancel the common factor of $13$ .

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Step 3.4.2.3.1.1.1.2.1

Cancel the common factor.

$u^{\frac{1}{13} \cdot 13} = 2^{13}$

Step 3.4.2.3.1.1.1.2.2

Rewrite the expression.

$u^{1} = 2^{13}$

Step 3.4.2.3.1.1.2

Simplify.

$u = 2^{13}$

Step 3.4.2.3.2

Simplify the right side.

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Step 3.4.2.3.2.1

Raise $2$ to the power of $13$ .

$u = 8192$

Step 3.5

Set $2 u^{\frac{1}{13}} + u^{\frac{2}{13}} + 4$ equal to $0$ and solve for $u$ .

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Step 3.5.1

Set $2 u^{\frac{1}{13}} + u^{\frac{2}{13}} + 4$ equal to $0$ .

$2 u^{\frac{1}{13}} + u^{\frac{2}{13}} + 4 = 0$

Step 3.5.2

Solve $2 u^{\frac{1}{13}} + u^{\frac{2}{13}} + 4 = 0$ for $u$ .

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Step 3.5.2.1

Find a common factor $u^{\frac{1}{13}}$ that is present in each term.

$2 u^{\frac{1}{13}} {(u^{\frac{1}{13}})}^{2} + 4$

Step 3.5.2.2

Substitute $u$ for $u^{\frac{1}{13}}$ .

$2 (u) {(u)}^{2} + 4 = 0$

Step 3.5.2.3

Solve for $u$ .

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Step 3.5.2.3.1

Multiply $u$ by ${(u)}^{2}$ by adding the exponents.

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Step 3.5.2.3.1.1

Move ${(u)}^{2}$ .

$2 ({(u)}^{2} u) + 4 = 0$

Step 3.5.2.3.1.2

Multiply ${(u)}^{2}$ by $u$ .

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Step 3.5.2.3.1.2.1

Raise $u$ to the power of $1$ .

$2 ({(u)}^{2} u^{1}) + 4 = 0$

Step 3.5.2.3.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 u^{2 + 1} + 4 = 0$

Step 3.5.2.3.1.3

Add $2$ and $1$ .

$2 u^{3} + 4 = 0$

Step 3.5.2.3.2

Subtract $4$ from both sides of the equation.

$2 u^{3} = - 4$

Step 3.5.2.3.3

Add $4$ to both sides of the equation.

$2 u^{3} + 4 = 0$

Step 3.5.2.3.4

Factor $2$ out of $2 u^{3} + 4$ .

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Step 3.5.2.3.4.1

Factor $2$ out of $2 u^{3}$ .

$2 (u^{3}) + 4 = 0$

Step 3.5.2.3.4.2

Factor $2$ out of $4$ .

$2 u^{3} + 2 \cdot 2 = 0$

Step 3.5.2.3.4.3

Factor $2$ out of $2 u^{3} + 2 \cdot 2$ .

$2 (u^{3} + 2) = 0$

Step 3.5.2.3.5

Divide each term in $2 (u^{3} + 2) = 0$ by $2$ and simplify.

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Step 3.5.2.3.5.1

Divide each term in $2 (u^{3} + 2) = 0$ by $2$ .

$\frac{2 (u^{3} + 2)}{2} = \frac{0}{2}$

Step 3.5.2.3.5.2

Simplify the left side.

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Step 3.5.2.3.5.2.1

Cancel the common factor of $2$ .

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Step 3.5.2.3.5.2.1.1

Cancel the common factor.

$\frac{2 (u^{3} + 2)}{2} = \frac{0}{2}$

Step 3.5.2.3.5.2.1.2

Divide $u^{3} + 2$ by $1$ .

$u^{3} + 2 = \frac{0}{2}$

Step 3.5.2.3.5.3

Simplify the right side.

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Step 3.5.2.3.5.3.1

Divide $0$ by $2$ .

$u^{3} + 2 = 0$

Step 3.5.2.3.6

Subtract $2$ from both sides of the equation.

$u^{3} = - 2$

Step 3.5.2.3.7

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$u = \sqrt[3]{- 2}$

Step 3.5.2.3.8

Simplify $\sqrt[3]{- 2}$ .

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Step 3.5.2.3.8.1

Rewrite $- 2$ as ${(- 1)}^{3} \cdot 2$ .

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Step 3.5.2.3.8.1.1

Rewrite $- 2$ as $- 1 (2)$ .

$u = \sqrt[3]{- 1 (2)}$

Step 3.5.2.3.8.1.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$u = \sqrt[3]{{(- 1)}^{3} \cdot 2}$

Step 3.5.2.3.8.2

Pull terms out from under the radical.

$u = - 1 \sqrt[3]{2}$

Step 3.5.2.3.8.3

Rewrite $- 1 \sqrt[3]{2}$ as $- \sqrt[3]{2}$ .

$u = - \sqrt[3]{2}$

Step 3.5.2.4

Substitute $u$ for $u$ .

$u^{\frac{1}{13}} = - \sqrt[3]{2}$

Step 3.5.2.5

Solve for $u$ .

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Step 3.5.2.5.1

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

${(u^{\frac{1}{13}})}^{13} = {(- \sqrt[3]{2})}^{13}$

Step 3.5.2.5.2

Simplify the exponent.

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Step 3.5.2.5.2.1

Simplify the left side.

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Step 3.5.2.5.2.1.1

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

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Step 3.5.2.5.2.1.1.1

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

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Step 3.5.2.5.2.1.1.1.1

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

$u^{\frac{1}{13} \cdot 13} = {(- \sqrt[3]{2})}^{13}$

Step 3.5.2.5.2.1.1.1.2

Cancel the common factor of $13$ .

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Step 3.5.2.5.2.1.1.1.2.1

Cancel the common factor.

$u^{\frac{1}{13} \cdot 13} = {(- \sqrt[3]{2})}^{13}$

Step 3.5.2.5.2.1.1.1.2.2

Rewrite the expression.

$u^{1} = {(- \sqrt[3]{2})}^{13}$

Step 3.5.2.5.2.1.1.2

Simplify.

$u = {(- \sqrt[3]{2})}^{13}$

Step 3.5.2.5.2.2

Simplify the right side.

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Step 3.5.2.5.2.2.1

Simplify ${(- \sqrt[3]{2})}^{13}$ .

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Step 3.5.2.5.2.2.1.1

Apply the product rule to $- \sqrt[3]{2}$ .

$u = {(- 1)}^{13} {\sqrt[3]{2}}^{13}$

Step 3.5.2.5.2.2.1.2

Raise $- 1$ to the power of $13$ .

$u = - {\sqrt[3]{2}}^{13}$

Step 3.5.2.5.2.2.1.3

Rewrite ${\sqrt[3]{2}}^{13}$ as $\sqrt[3]{2^{13}}$ .

$u = - \sqrt[3]{2^{13}}$

Step 3.5.2.5.2.2.1.4

Raise $2$ to the power of $13$ .

$u = - \sqrt[3]{8192}$

Step 3.5.2.5.2.2.1.5

Rewrite $8192$ as $16^{3} \cdot 2$ .

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Step 3.5.2.5.2.2.1.5.1

Factor $4096$ out of $8192$ .

$u = - \sqrt[3]{4096 (2)}$

Step 3.5.2.5.2.2.1.5.2

Rewrite $4096$ as $16^{3}$ .

$u = - \sqrt[3]{16^{3} \cdot 2}$

Step 3.5.2.5.2.2.1.6

Pull terms out from under the radical.

$u = - (16 \sqrt[3]{2})$

Step 3.5.2.5.2.2.1.7

Multiply $16$ by $- 1$ .

$u = - 16 \sqrt[3]{2}$

Step 3.6

The final solution is all the values that make $2 u (u^{\frac{1}{13}} - 2) (2 u^{\frac{1}{13}} + u^{\frac{2}{13}} + 4) = 0$ true.

$u = 0, 8192, - 16 \sqrt[3]{2}$

Step 4

Substitute $x$ for $u$ .

$x^{\frac{13}{5}} = 0, 8192, - 16 \sqrt[3]{2}$

Step 5

Solve for $x^{\frac{13}{5}} = 0$ for $x$ .

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Step 5.1

Raise each side of the equation to the power of $\frac{5}{13}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{13}{5}})}^{\frac{5}{13}} = 0^{\frac{5}{13}}$

Step 5.2

Simplify the exponent.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Simplify ${(x^{\frac{13}{5}})}^{\frac{5}{13}}$ .

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Step 5.2.1.1.1

Multiply the exponents in ${(x^{\frac{13}{5}})}^{\frac{5}{13}}$ .

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Step 5.2.1.1.1.1

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

$x^{\frac{13}{5} \cdot \frac{5}{13}} = 0^{\frac{5}{13}}$

Step 5.2.1.1.1.2

Cancel the common factor of $13$ .

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Step 5.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{13}{5} \cdot \frac{5}{13}} = 0^{\frac{5}{13}}$

Step 5.2.1.1.1.2.2

Rewrite the expression.

$x^{\frac{1}{5} \cdot 5} = 0^{\frac{5}{13}}$

Step 5.2.1.1.1.3

Cancel the common factor of $5$ .

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Step 5.2.1.1.1.3.1

Cancel the common factor.

$x^{\frac{1}{5} \cdot 5} = 0^{\frac{5}{13}}$

Step 5.2.1.1.1.3.2

Rewrite the expression.

$x^{1} = 0^{\frac{5}{13}}$

Step 5.2.1.1.2

Simplify.

$x = 0^{\frac{5}{13}}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $0^{\frac{5}{13}}$ .

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Step 5.2.2.1.1

Simplify the expression.

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Step 5.2.2.1.1.1

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

$x = {(0^{13})}^{\frac{5}{13}}$

Step 5.2.2.1.1.2

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

$x = 0^{13 (\frac{5}{13})}$

Step 5.2.2.1.2

Cancel the common factor of $13$ .

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Step 5.2.2.1.2.1

Cancel the common factor.

$x = 0^{13 (\frac{5}{13})}$

Step 5.2.2.1.2.2

Rewrite the expression.

$x = 0^{5}$

Step 5.2.2.1.3

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

$x = 0$

Step 6

Solve for $x^{\frac{13}{5}} = 8192$ for $x$ .

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Step 6.1

Raise each side of the equation to the power of $\frac{5}{13}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{13}{5}})}^{\frac{5}{13}} = 8192^{\frac{5}{13}}$

Step 6.2

Simplify the exponent.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Simplify ${(x^{\frac{13}{5}})}^{\frac{5}{13}}$ .

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Step 6.2.1.1.1

Multiply the exponents in ${(x^{\frac{13}{5}})}^{\frac{5}{13}}$ .

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Step 6.2.1.1.1.1

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

$x^{\frac{13}{5} \cdot \frac{5}{13}} = 8192^{\frac{5}{13}}$

Step 6.2.1.1.1.2

Cancel the common factor of $13$ .

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Step 6.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{13}{5} \cdot \frac{5}{13}} = 8192^{\frac{5}{13}}$

Step 6.2.1.1.1.2.2

Rewrite the expression.

$x^{\frac{1}{5} \cdot 5} = 8192^{\frac{5}{13}}$

Step 6.2.1.1.1.3

Cancel the common factor of $5$ .

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Step 6.2.1.1.1.3.1

Cancel the common factor.

$x^{\frac{1}{5} \cdot 5} = 8192^{\frac{5}{13}}$

Step 6.2.1.1.1.3.2

Rewrite the expression.

$x^{1} = 8192^{\frac{5}{13}}$

Step 6.2.1.1.2

Simplify.

$x = 8192^{\frac{5}{13}}$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify $8192^{\frac{5}{13}}$ .

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Step 6.2.2.1.1

Simplify the expression.

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Step 6.2.2.1.1.1

Rewrite $8192$ as $2^{13}$ .

$x = {(2^{13})}^{\frac{5}{13}}$

Step 6.2.2.1.1.2

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

$x = 2^{13 (\frac{5}{13})}$

Step 6.2.2.1.2

Cancel the common factor of $13$ .

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Step 6.2.2.1.2.1

Cancel the common factor.

$x = 2^{13 (\frac{5}{13})}$

Step 6.2.2.1.2.2

Rewrite the expression.

$x = 2^{5}$

Step 6.2.2.1.3

Raise $2$ to the power of $5$ .

$x = 32$

Step 7

Solve for $x^{\frac{13}{5}} = - 16 \sqrt[3]{2}$ for $x$ .

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Step 7.1

Raise each side of the equation to the power of $\frac{5}{13}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{13}{5}})}^{\frac{5}{13}} = {(- 16 \sqrt[3]{2})}^{\frac{5}{13}}$

Step 7.2

Simplify the exponent.

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Step 7.2.1

Simplify the left side.

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Step 7.2.1.1

Simplify ${(x^{\frac{13}{5}})}^{\frac{5}{13}}$ .

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Step 7.2.1.1.1

Multiply the exponents in ${(x^{\frac{13}{5}})}^{\frac{5}{13}}$ .

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Step 7.2.1.1.1.1

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

$x^{\frac{13}{5} \cdot \frac{5}{13}} = {(- 16 \sqrt[3]{2})}^{\frac{5}{13}}$

Step 7.2.1.1.1.2

Cancel the common factor of $13$ .

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Step 7.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{13}{5} \cdot \frac{5}{13}} = {(- 16 \sqrt[3]{2})}^{\frac{5}{13}}$

Step 7.2.1.1.1.2.2

Rewrite the expression.

$x^{\frac{1}{5} \cdot 5} = {(- 16 \sqrt[3]{2})}^{\frac{5}{13}}$

Step 7.2.1.1.1.3

Cancel the common factor of $5$ .

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Step 7.2.1.1.1.3.1

Cancel the common factor.

$x^{\frac{1}{5} \cdot 5} = {(- 16 \sqrt[3]{2})}^{\frac{5}{13}}$

Step 7.2.1.1.1.3.2

Rewrite the expression.

$x^{1} = {(- 16 \sqrt[3]{2})}^{\frac{5}{13}}$

Step 7.2.1.1.2

Simplify.

$x = {(- 16 \sqrt[3]{2})}^{\frac{5}{13}}$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Apply the product rule to $- 16 \sqrt[3]{2}$ .

$x = {(- 16)}^{\frac{5}{13}} {\sqrt[3]{2}}^{\frac{5}{13}}$

Step 8

List all of the solutions.

$x = 0, 32, {(- 16)}^{\frac{5}{13}} {\sqrt[3]{2}}^{\frac{5}{13}}$

Step 9

Exclude the solutions that do not make $2 x^{\frac{16}{5}} - 16 x^{\frac{13}{5}} = 0$ true.

$x = 0, 32$