Algebra Examples

$x^{4} - 128 x^{\frac{13}{5}} = 0$

Step 1

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

${(x^{\frac{13}{5}})}^{\frac{20}{13}} - 128 x^{\frac{13}{5}}$

Step 2

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

${(u)}^{\frac{20}{13}} - 128 u = 0$

Step 3

Solve for $u$ .

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Step 3.1

Factor $u$ out of $u^{\frac{20}{13}} - 128 u$ .

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Step 3.1.1

Factor $u$ out of $u^{\frac{20}{13}}$ .

$u \cdot u^{\frac{7}{13}} - 128 u = 0$

Step 3.1.2

Factor $u$ out of $- 128 u$ .

$u \cdot u^{\frac{7}{13}} + u \cdot - 128 = 0$

Step 3.1.3

Factor $u$ out of $u \cdot u^{\frac{7}{13}} + u \cdot - 128$ .

$u (u^{\frac{7}{13}} - 128) = 0$

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 $0$ .

$u = 0$

$u^{\frac{7}{13}} - 128 = 0$

Step 3.3

Set $u$ equal to $0$ .

$u = 0$

Step 3.4

Set $u^{\frac{7}{13}} - 128$ equal to $0$ and solve for $u$ .

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Step 3.4.1

Set $u^{\frac{7}{13}} - 128$ equal to $0$ .

$u^{\frac{7}{13}} - 128 = 0$

Step 3.4.2

Solve $u^{\frac{7}{13}} - 128 = 0$ for $u$ .

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Step 3.4.2.1

Add $128$ to both sides of the equation.

$u^{\frac{7}{13}} = 128$

Step 3.4.2.2

Raise each side of the equation to the power of $\frac{13}{7}$ to eliminate the fractional exponent on the left side.

${(u^{\frac{7}{13}})}^{\frac{13}{7}} = 128^{\frac{13}{7}}$

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 ${(u^{\frac{7}{13}})}^{\frac{13}{7}}$ .

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Step 3.4.2.3.1.1.1

Multiply the exponents in ${(u^{\frac{7}{13}})}^{\frac{13}{7}}$ .

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Step 3.4.2.3.1.1.1.1

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

$u^{\frac{7}{13} \cdot \frac{13}{7}} = 128^{\frac{13}{7}}$

Step 3.4.2.3.1.1.1.2

Cancel the common factor of $7$ .

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Step 3.4.2.3.1.1.1.2.1

Cancel the common factor.

$u^{\frac{7}{13} \cdot \frac{13}{7}} = 128^{\frac{13}{7}}$

Step 3.4.2.3.1.1.1.2.2

Rewrite the expression.

$u^{\frac{1}{13} \cdot 13} = 128^{\frac{13}{7}}$

Step 3.4.2.3.1.1.1.3

Cancel the common factor of $13$ .

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Step 3.4.2.3.1.1.1.3.1

Cancel the common factor.

$u^{\frac{1}{13} \cdot 13} = 128^{\frac{13}{7}}$

Step 3.4.2.3.1.1.1.3.2

Rewrite the expression.

$u^{1} = 128^{\frac{13}{7}}$

Step 3.4.2.3.1.1.2

Simplify.

$u = 128^{\frac{13}{7}}$

Step 3.4.2.3.2

Simplify the right side.

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Step 3.4.2.3.2.1

Simplify $128^{\frac{13}{7}}$ .

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Step 3.4.2.3.2.1.1

Simplify the expression.

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Step 3.4.2.3.2.1.1.1

Rewrite $128$ as $2^{7}$ .

$u = {(2^{7})}^{\frac{13}{7}}$

Step 3.4.2.3.2.1.1.2

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

$u = 2^{7 (\frac{13}{7})}$

Step 3.4.2.3.2.1.2

Cancel the common factor of $7$ .

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Step 3.4.2.3.2.1.2.1

Cancel the common factor.

$u = 2^{7 (\frac{13}{7})}$

Step 3.4.2.3.2.1.2.2

Rewrite the expression.

$u = 2^{13}$

Step 3.4.2.3.2.1.3

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

$u = 8192$

Step 3.5

The final solution is all the values that make $u (u^{\frac{7}{13}} - 128) = 0$ true.

$u = 0, 8192$

Step 4

Substitute $x$ for $u$ .

$x^{\frac{13}{5}} = 0, 8192$

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

List all of the solutions.

$x = 0, 32$