Algebra Examples

$f (x) = \frac{1}{x^{4}} - 7$

Step 1

Write $f (x) = \frac{1}{x^{4}} - 7$ as an equation.

$y = \frac{1}{x^{4}} - 7$

Step 2

Interchange the variables.

$x = \frac{1}{y^{4}} - 7$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{1}{y^{4}} - 7 = x$ .

$\frac{1}{y^{4}} - 7 = x$

Step 3.2

Add $7$ to both sides of the equation.

$\frac{1}{y^{4}} = x + 7$

Step 3.3

Find the LCD of the terms in the equation.

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Step 3.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y^{4}, 1, 1$

Step 3.3.2

The LCM of one and any expression is the expression.

$y^{4}$

Step 3.4

Multiply each term in $\frac{1}{y^{4}} = x + 7$ by $y^{4}$ to eliminate the fractions.

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Step 3.4.1

Multiply each term in $\frac{1}{y^{4}} = x + 7$ by $y^{4}$ .

$\frac{1}{y^{4}} y^{4} = x y^{4} + 7 y^{4}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $y^{4}$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{1}{y^{4}} y^{4} = x y^{4} + 7 y^{4}$

Step 3.4.2.1.2

Rewrite the expression.

$1 = x y^{4} + 7 y^{4}$

Step 3.5

Solve the equation.

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Step 3.5.1

Rewrite the equation as $x y^{4} + 7 y^{4} = 1$ .

$x y^{4} + 7 y^{4} = 1$

Step 3.5.2

Factor $y^{4}$ out of $x y^{4} + 7 y^{4}$ .

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Step 3.5.2.1

Factor $y^{4}$ out of $x y^{4}$ .

$y^{4} x + 7 y^{4} = 1$

Step 3.5.2.2

Factor $y^{4}$ out of $7 y^{4}$ .

$y^{4} x + y^{4} \cdot 7 = 1$

Step 3.5.2.3

Factor $y^{4}$ out of $y^{4} x + y^{4} \cdot 7$ .

$y^{4} (x + 7) = 1$

Step 3.5.3

Divide each term in $y^{4} (x + 7) = 1$ by $x + 7$ and simplify.

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Step 3.5.3.1

Divide each term in $y^{4} (x + 7) = 1$ by $x + 7$ .

$\frac{y^{4} (x + 7)}{x + 7} = \frac{1}{x + 7}$

Step 3.5.3.2

Simplify the left side.

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Step 3.5.3.2.1

Cancel the common factor of $x + 7$ .

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Step 3.5.3.2.1.1

Cancel the common factor.

$\frac{y^{4} (x + 7)}{x + 7} = \frac{1}{x + 7}$

Step 3.5.3.2.1.2

Divide $y^{4}$ by $1$ .

$y^{4} = \frac{1}{x + 7}$

Step 3.5.4

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

$y = \pm \sqrt[4]{\frac{1}{x + 7}}$

Step 3.5.5

Simplify $\pm \sqrt[4]{\frac{1}{x + 7}}$ .

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Step 3.5.5.1

Rewrite $\sqrt[4]{\frac{1}{x + 7}}$ as $\frac{\sqrt[4]{1}}{\sqrt[4]{x + 7}}$ .

$y = \pm \frac{\sqrt[4]{1}}{\sqrt[4]{x + 7}}$

Step 3.5.5.2

Any root of $1$ is $1$ .

$y = \pm \frac{1}{\sqrt[4]{x + 7}}$

Step 3.5.5.3

Multiply $\frac{1}{\sqrt[4]{x + 7}}$ by $\frac{{\sqrt[4]{x + 7}}^{3}}{{\sqrt[4]{x + 7}}^{3}}$ .

$y = \pm \frac{1}{\sqrt[4]{x + 7}} \cdot \frac{{\sqrt[4]{x + 7}}^{3}}{{\sqrt[4]{x + 7}}^{3}}$

Step 3.5.5.4

Combine and simplify the denominator.

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Step 3.5.5.4.1

Multiply $\frac{1}{\sqrt[4]{x + 7}}$ by $\frac{{\sqrt[4]{x + 7}}^{3}}{{\sqrt[4]{x + 7}}^{3}}$ .

$y = \pm \frac{{\sqrt[4]{x + 7}}^{3}}{\sqrt[4]{x + 7} {\sqrt[4]{x + 7}}^{3}}$

Step 3.5.5.4.2

Raise $\sqrt[4]{x + 7}$ to the power of $1$ .

$y = \pm \frac{{\sqrt[4]{x + 7}}^{3}}{{\sqrt[4]{x + 7}}^{1} {\sqrt[4]{x + 7}}^{3}}$

Step 3.5.5.4.3

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

$y = \pm \frac{{\sqrt[4]{x + 7}}^{3}}{{\sqrt[4]{x + 7}}^{1 + 3}}$

Step 3.5.5.4.4

Add $1$ and $3$ .

$y = \pm \frac{{\sqrt[4]{x + 7}}^{3}}{{\sqrt[4]{x + 7}}^{4}}$

Step 3.5.5.4.5

Rewrite ${\sqrt[4]{x + 7}}^{4}$ as $x + 7$ .

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Step 3.5.5.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x + 7}$ as ${(x + 7)}^{\frac{1}{4}}$ .

$y = \pm \frac{{\sqrt[4]{x + 7}}^{3}}{{({(x + 7)}^{\frac{1}{4}})}^{4}}$

Step 3.5.5.4.5.2

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

$y = \pm \frac{{\sqrt[4]{x + 7}}^{3}}{{(x + 7)}^{\frac{1}{4} \cdot 4}}$

Step 3.5.5.4.5.3

Combine $\frac{1}{4}$ and $4$ .

$y = \pm \frac{{\sqrt[4]{x + 7}}^{3}}{{(x + 7)}^{\frac{4}{4}}}$

Step 3.5.5.4.5.4

Cancel the common factor of $4$ .

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Step 3.5.5.4.5.4.1

Cancel the common factor.

$y = \pm \frac{{\sqrt[4]{x + 7}}^{3}}{{(x + 7)}^{\frac{4}{4}}}$

Step 3.5.5.4.5.4.2

Rewrite the expression.

$y = \pm \frac{{\sqrt[4]{x + 7}}^{3}}{{(x + 7)}^{1}}$

Step 3.5.5.4.5.5

Simplify.

$y = \pm \frac{{\sqrt[4]{x + 7}}^{3}}{x + 7}$

Step 3.5.5.5

Rewrite ${\sqrt[4]{x + 7}}^{3}$ as $\sqrt[4]{{(x + 7)}^{3}}$ .

$y = \pm \frac{\sqrt[4]{{(x + 7)}^{3}}}{x + 7}$

Step 3.5.6

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

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Step 3.5.6.1

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

$y = \frac{\sqrt[4]{{(x + 7)}^{3}}}{x + 7}$

Step 3.5.6.2

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