Algebra Examples

$\sqrt[5]{x^{4}} - 3 \sqrt[5]{x^{2}} = 4$

Step 1

Simplify the left side.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{4}}$ as $x^{\frac{4}{5}}$ .

$x^{\frac{4}{5}} - 3 \sqrt[5]{x^{2}} = 4$

Step 1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{2}}$ as $x^{\frac{2}{5}}$ .

$x^{\frac{4}{5}} - 3 x^{\frac{2}{5}} = 4$

Step 2

Find a common factor $x^{\frac{2}{5}}$ that is present in each term.

${(x^{\frac{2}{5}})}^{2} - 3 x^{\frac{2}{5}} - 4$

Step 3

Substitute $u$ for $x^{\frac{2}{5}}$ .

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

Step 4

Solve for $u$ .

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Step 4.1

Remove parentheses.

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

Step 4.2

Factor $u^{2} - 3 u - 4$ using the AC method.

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Step 4.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 4$ and whose sum is $- 3$ .

$- 4, 1$

Step 4.2.2

Write the factored form using these integers.

$(u - 4) (u + 1) = 0$

Step 4.3

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

$u - 4 = 0$

$u + 1 = 0$

Step 4.4

Set $u - 4$ equal to $0$ and solve for $u$ .

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Step 4.4.1

Set $u - 4$ equal to $0$ .

$u - 4 = 0$

Step 4.4.2

Add $4$ to both sides of the equation.

$u = 4$

Step 4.5

Set $u + 1$ equal to $0$ and solve for $u$ .

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Step 4.5.1

Set $u + 1$ equal to $0$ .

$u + 1 = 0$

Step 4.5.2

Subtract $1$ from both sides of the equation.

$u = - 1$

Step 4.6

The final solution is all the values that make $(u - 4) (u + 1) = 0$ true.

$u = 4, - 1$

Step 5

Substitute $x$ for $u$ .

$x^{\frac{2}{5}} = 4, - 1$

Step 6

Solve for $x^{\frac{2}{5}} = 4$ for $x$ .

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Step 6.1

Raise each side of the equation to the power of $\frac{5}{2}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{2}{5}})}^{\frac{5}{2}} = \pm 4^{\frac{5}{2}}$

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 ${(x^{\frac{2}{5}})}^{\frac{5}{2}}$ .

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Step 6.2.1.1.1

Multiply the exponents in ${(x^{\frac{2}{5}})}^{\frac{5}{2}}$ .

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Step 6.2.1.1.1.1

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

$x^{\frac{2}{5} \cdot \frac{5}{2}} = \pm 4^{\frac{5}{2}}$

Step 6.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 6.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{2}{5} \cdot \frac{5}{2}} = \pm 4^{\frac{5}{2}}$

Step 6.2.1.1.1.2.2

Rewrite the expression.

$x^{\frac{1}{5} \cdot 5} = \pm 4^{\frac{5}{2}}$

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.

$x^{\frac{1}{5} \cdot 5} = \pm 4^{\frac{5}{2}}$

Step 6.2.1.1.1.3.2

Rewrite the expression.

$x^{1} = \pm 4^{\frac{5}{2}}$

Step 6.2.1.1.2

Simplify.

$x = \pm 4^{\frac{5}{2}}$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify $\pm 4^{\frac{5}{2}}$ .

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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 $4$ as $2^{2}$ .

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

Step 6.2.2.1.1.2

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

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

Step 6.2.2.1.2

Cancel the common factor of $2$ .

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Step 6.2.2.1.2.1

Cancel the common factor.

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

Step 6.2.2.1.2.2

Rewrite the expression.

$x = \pm 2^{5}$

Step 6.2.2.1.3

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

$x = \pm 32$

Step 6.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 32$

Step 6.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 32$

Step 6.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 32, - 32$

Step 7

Solve for $x^{\frac{2}{5}} = - 1$ for $x$ .

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Step 7.1

Raise each side of the equation to the power of $\frac{5}{2}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{2}{5}})}^{\frac{5}{2}} = \pm {(- 1)}^{\frac{5}{2}}$

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 ${(x^{\frac{2}{5}})}^{\frac{5}{2}}$ .

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Step 7.2.1.1.1

Multiply the exponents in ${(x^{\frac{2}{5}})}^{\frac{5}{2}}$ .

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Step 7.2.1.1.1.1

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

$x^{\frac{2}{5} \cdot \frac{5}{2}} = \pm {(- 1)}^{\frac{5}{2}}$

Step 7.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 7.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{2}{5} \cdot \frac{5}{2}} = \pm {(- 1)}^{\frac{5}{2}}$

Step 7.2.1.1.1.2.2

Rewrite the expression.

$x^{\frac{1}{5} \cdot 5} = \pm {(- 1)}^{\frac{5}{2}}$

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.

$x^{\frac{1}{5} \cdot 5} = \pm {(- 1)}^{\frac{5}{2}}$

Step 7.2.1.1.1.3.2

Rewrite the expression.

$x^{1} = \pm {(- 1)}^{\frac{5}{2}}$

Step 7.2.1.1.2

Simplify.

$x = \pm {(- 1)}^{\frac{5}{2}}$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $\pm {(- 1)}^{\frac{5}{2}}$ .

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Step 7.2.2.1.1

Rewrite ${(- 1)}^{\frac{5}{2}}$ as $\sqrt{{(- 1)}^{5}}$ .

$x = \pm \sqrt{{(- 1)}^{5}}$

Step 7.2.2.1.2

Raise $- 1$ to the power of $5$ .

$x = \pm \sqrt{- 1}$

Step 7.2.2.1.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i$

Step 7.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 7.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i$

Step 7.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i$

Step 7.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i, - i$

Step 8

List all of the solutions.

$x = 32, - 32, i, - i$