Linear Algebra Examples

$9 x^{4} + 10 y^{4} = 42$

Step 1

Subtract $9 x^{4}$ from both sides of the equation.

$10 y^{4} = 42 - 9 x^{4}$

Step 2

Divide each term in $10 y^{4} = 42 - 9 x^{4}$ by $10$ and simplify.

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Step 2.1

Divide each term in $10 y^{4} = 42 - 9 x^{4}$ by $10$ .

$\frac{10 y^{4}}{10} = \frac{42}{10} + \frac{- 9 x^{4}}{10}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $10$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{10 y^{4}}{10} = \frac{42}{10} + \frac{- 9 x^{4}}{10}$

Step 2.2.1.2

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

$y^{4} = \frac{42}{10} + \frac{- 9 x^{4}}{10}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Cancel the common factor of $42$ and $10$ .

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Step 2.3.1.1.1

Factor $2$ out of $42$ .

$y^{4} = \frac{2 (21)}{10} + \frac{- 9 x^{4}}{10}$

Step 2.3.1.1.2

Cancel the common factors.

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Step 2.3.1.1.2.1

Factor $2$ out of $10$ .

$y^{4} = \frac{2 \cdot 21}{2 \cdot 5} + \frac{- 9 x^{4}}{10}$

Step 2.3.1.1.2.2

Cancel the common factor.

$y^{4} = \frac{2 \cdot 21}{2 \cdot 5} + \frac{- 9 x^{4}}{10}$

Step 2.3.1.1.2.3

Rewrite the expression.

$y^{4} = \frac{21}{5} + \frac{- 9 x^{4}}{10}$

Step 2.3.1.2

Move the negative in front of the fraction.

$y^{4} = \frac{21}{5} - \frac{9 x^{4}}{10}$

Step 3

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

$y = \pm \sqrt[4]{\frac{21}{5} - \frac{9 x^{4}}{10}}$

Step 4

Simplify $\pm \sqrt[4]{\frac{21}{5} - \frac{9 x^{4}}{10}}$ .

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Step 4.1

To write $\frac{21}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \pm \sqrt[4]{\frac{21}{5} \cdot \frac{2}{2} - \frac{9 x^{4}}{10}}$

Step 4.2

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.1

Multiply $\frac{21}{5}$ by $\frac{2}{2}$ .

$y = \pm \sqrt[4]{\frac{21 \cdot 2}{5 \cdot 2} - \frac{9 x^{4}}{10}}$

Step 4.2.2

Multiply $5$ by $2$ .

$y = \pm \sqrt[4]{\frac{21 \cdot 2}{10} - \frac{9 x^{4}}{10}}$

Step 4.3

Combine the numerators over the common denominator.

$y = \pm \sqrt[4]{\frac{21 \cdot 2 - 9 x^{4}}{10}}$

Step 4.4

Simplify the numerator.

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Step 4.4.1

Factor $3$ out of $21 \cdot 2 - 9 x^{4}$ .

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Step 4.4.1.1

Factor $3$ out of $21 \cdot 2$ .

$y = \pm \sqrt[4]{\frac{3 (7 \cdot 2) - 9 x^{4}}{10}}$

Step 4.4.1.2

Factor $3$ out of $- 9 x^{4}$ .

$y = \pm \sqrt[4]{\frac{3 (7 \cdot 2) + 3 (- 3 x^{4})}{10}}$

Step 4.4.1.3

Factor $3$ out of $3 (7 \cdot 2) + 3 (- 3 x^{4})$ .

$y = \pm \sqrt[4]{\frac{3 (7 \cdot 2 - 3 x^{4})}{10}}$

Step 4.4.2

Multiply $7$ by $2$ .

$y = \pm \sqrt[4]{\frac{3 (14 - 3 x^{4})}{10}}$

Step 4.5

Rewrite $\sqrt[4]{\frac{3 (14 - 3 x^{4})}{10}}$ as $\frac{\sqrt[4]{3 (14 - 3 x^{4})}}{\sqrt[4]{10}}$ .

$y = \pm \frac{\sqrt[4]{3 (14 - 3 x^{4})}}{\sqrt[4]{10}}$

Step 4.6

Multiply $\frac{\sqrt[4]{3 (14 - 3 x^{4})}}{\sqrt[4]{10}}$ by $\frac{{\sqrt[4]{10}}^{3}}{{\sqrt[4]{10}}^{3}}$ .

$y = \pm \frac{\sqrt[4]{3 (14 - 3 x^{4})}}{\sqrt[4]{10}} \cdot \frac{{\sqrt[4]{10}}^{3}}{{\sqrt[4]{10}}^{3}}$

Step 4.7

Combine and simplify the denominator.

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Step 4.7.1

Multiply $\frac{\sqrt[4]{3 (14 - 3 x^{4})}}{\sqrt[4]{10}}$ by $\frac{{\sqrt[4]{10}}^{3}}{{\sqrt[4]{10}}^{3}}$ .

$y = \pm \frac{\sqrt[4]{3 (14 - 3 x^{4})} {\sqrt[4]{10}}^{3}}{\sqrt[4]{10} {\sqrt[4]{10}}^{3}}$

Step 4.7.2

Raise $\sqrt[4]{10}$ to the power of $1$ .

$y = \pm \frac{\sqrt[4]{3 (14 - 3 x^{4})} {\sqrt[4]{10}}^{3}}{{\sqrt[4]{10}}^{1} {\sqrt[4]{10}}^{3}}$

Step 4.7.3

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

$y = \pm \frac{\sqrt[4]{3 (14 - 3 x^{4})} {\sqrt[4]{10}}^{3}}{{\sqrt[4]{10}}^{1 + 3}}$

Step 4.7.4

Add $1$ and $3$ .

$y = \pm \frac{\sqrt[4]{3 (14 - 3 x^{4})} {\sqrt[4]{10}}^{3}}{{\sqrt[4]{10}}^{4}}$

Step 4.7.5

Rewrite ${\sqrt[4]{10}}^{4}$ as $10$ .

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Step 4.7.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{10}$ as $10^{\frac{1}{4}}$ .

$y = \pm \frac{\sqrt[4]{3 (14 - 3 x^{4})} {\sqrt[4]{10}}^{3}}{{(10^{\frac{1}{4}})}^{4}}$

Step 4.7.5.2

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

$y = \pm \frac{\sqrt[4]{3 (14 - 3 x^{4})} {\sqrt[4]{10}}^{3}}{10^{\frac{1}{4} \cdot 4}}$

Step 4.7.5.3

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

$y = \pm \frac{\sqrt[4]{3 (14 - 3 x^{4})} {\sqrt[4]{10}}^{3}}{10^{\frac{4}{4}}}$

Step 4.7.5.4

Cancel the common factor of $4$ .

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Step 4.7.5.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt[4]{3 (14 - 3 x^{4})} {\sqrt[4]{10}}^{3}}{10^{\frac{4}{4}}}$

Step 4.7.5.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt[4]{3 (14 - 3 x^{4})} {\sqrt[4]{10}}^{3}}{10^{1}}$

Step 4.7.5.5

Evaluate the exponent.

$y = \pm \frac{\sqrt[4]{3 (14 - 3 x^{4})} {\sqrt[4]{10}}^{3}}{10}$

Step 4.8

Simplify the numerator.

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Step 4.8.1

Rewrite ${\sqrt[4]{10}}^{3}$ as $\sqrt[4]{10^{3}}$ .

$y = \pm \frac{\sqrt[4]{3 (14 - 3 x^{4})} \sqrt[4]{10^{3}}}{10}$

Step 4.8.2

Raise $10$ to the power of $3$ .

$y = \pm \frac{\sqrt[4]{3 (14 - 3 x^{4})} \sqrt[4]{1000}}{10}$

Step 4.9

Simplify the numerator.

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Step 4.9.1

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt[4]{3 (14 - 3 x^{4}) \cdot 1000}}{10}$

Step 4.9.2

Multiply $1000$ by $3$ .

$y = \pm \frac{\sqrt[4]{3000 (14 - 3 x^{4})}}{10}$

Step 5

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

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Step 5.1

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

$y = \frac{\sqrt[4]{3000 (14 - 3 x^{4})}}{10}$

Step 5.2

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

$y = - \frac{\sqrt[4]{3000 (14 - 3 x^{4})}}{10}$

Step 5.3

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

$y = \frac{\sqrt[4]{3000 (14 - 3 x^{4})}}{10}$

$y = - \frac{\sqrt[4]{3000 (14 - 3 x^{4})}}{10}$

$y = \frac{\sqrt[4]{3000 (14 - 3 x^{4})}}{10}$

$y = - \frac{\sqrt[4]{3000 (14 - 3 x^{4})}}{10}$

Step 6

Set the radicand in $\sqrt[4]{3000 (14 - 3 x^{4})}$ greater than or equal to $0$ to find where the expression is defined.

$3000 (14 - 3 x^{4}) \geq 0$

Step 7

Solve for $x$ .

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Step 7.1

Divide each term in $3000 (14 - 3 x^{4}) \geq 0$ by $3000$ and simplify.

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Step 7.1.1

Divide each term in $3000 (14 - 3 x^{4}) \geq 0$ by $3000$ .

$\frac{3000 (14 - 3 x^{4})}{3000} \geq \frac{0}{3000}$

Step 7.1.2

Simplify the left side.

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Step 7.1.2.1

Cancel the common factor of $3000$ .

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Step 7.1.2.1.1

Cancel the common factor.

$\frac{3000 (14 - 3 x^{4})}{3000} \geq \frac{0}{3000}$

Step 7.1.2.1.2

Divide $14 - 3 x^{4}$ by $1$ .

$14 - 3 x^{4} \geq \frac{0}{3000}$

Step 7.1.3

Simplify the right side.

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Step 7.1.3.1

Divide $0$ by $3000$ .

$14 - 3 x^{4} \geq 0$

Step 7.2

Subtract $14$ from both sides of the inequality.

$- 3 x^{4} \geq - 14$

Step 7.3

Divide each term in $- 3 x^{4} \geq - 14$ by $- 3$ and simplify.

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Step 7.3.1

Divide each term in $- 3 x^{4} \geq - 14$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 x^{4}}{- 3} \leq \frac{- 14}{- 3}$

Step 7.3.2

Simplify the left side.

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Step 7.3.2.1

Cancel the common factor of $- 3$ .

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Step 7.3.2.1.1

Cancel the common factor.

$\frac{- 3 x^{4}}{- 3} \leq \frac{- 14}{- 3}$

Step 7.3.2.1.2

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

$x^{4} \leq \frac{- 14}{- 3}$

Step 7.3.3

Simplify the right side.

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Step 7.3.3.1

Dividing two negative values results in a positive value.

$x^{4} \leq \frac{14}{3}$

Step 7.4

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

$\sqrt[4]{x^{4}} \leq \sqrt[4]{\frac{14}{3}}$

Step 7.5

Simplify the equation.

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Step 7.5.1

Simplify the left side.

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Step 7.5.1.1

Pull terms out from under the radical.

$| x | \leq \sqrt[4]{\frac{14}{3}}$

Step 7.5.2

Simplify the right side.

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Step 7.5.2.1

Simplify $\sqrt[4]{\frac{14}{3}}$ .

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Step 7.5.2.1.1

Rewrite $\sqrt[4]{\frac{14}{3}}$ as $\frac{\sqrt[4]{14}}{\sqrt[4]{3}}$ .

$| x | \leq \frac{\sqrt[4]{14}}{\sqrt[4]{3}}$

Step 7.5.2.1.2

Multiply $\frac{\sqrt[4]{14}}{\sqrt[4]{3}}$ by $\frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{3}}$ .

$| x | \leq \frac{\sqrt[4]{14}}{\sqrt[4]{3}} \cdot \frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{3}}$

Step 7.5.2.1.3

Combine and simplify the denominator.

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Step 7.5.2.1.3.1

Multiply $\frac{\sqrt[4]{14}}{\sqrt[4]{3}}$ by $\frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{3}}$ .

$| x | \leq \frac{\sqrt[4]{14} {\sqrt[4]{3}}^{3}}{\sqrt[4]{3} {\sqrt[4]{3}}^{3}}$

Step 7.5.2.1.3.2

Raise $\sqrt[4]{3}$ to the power of $1$ .

$| x | \leq \frac{\sqrt[4]{14} {\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{1} {\sqrt[4]{3}}^{3}}$

Step 7.5.2.1.3.3

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

$| x | \leq \frac{\sqrt[4]{14} {\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{1 + 3}}$

Step 7.5.2.1.3.4

Add $1$ and $3$ .

$| x | \leq \frac{\sqrt[4]{14} {\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{4}}$

Step 7.5.2.1.3.5

Rewrite ${\sqrt[4]{3}}^{4}$ as $3$ .

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Step 7.5.2.1.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{3}$ as $3^{\frac{1}{4}}$ .

$| x | \leq \frac{\sqrt[4]{14} {\sqrt[4]{3}}^{3}}{{(3^{\frac{1}{4}})}^{4}}$

Step 7.5.2.1.3.5.2

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

$| x | \leq \frac{\sqrt[4]{14} {\sqrt[4]{3}}^{3}}{3^{\frac{1}{4} \cdot 4}}$

Step 7.5.2.1.3.5.3

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

$| x | \leq \frac{\sqrt[4]{14} {\sqrt[4]{3}}^{3}}{3^{\frac{4}{4}}}$

Step 7.5.2.1.3.5.4

Cancel the common factor of $4$ .

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Step 7.5.2.1.3.5.4.1

Cancel the common factor.

$| x | \leq \frac{\sqrt[4]{14} {\sqrt[4]{3}}^{3}}{3^{\frac{4}{4}}}$

Step 7.5.2.1.3.5.4.2

Rewrite the expression.

$| x | \leq \frac{\sqrt[4]{14} {\sqrt[4]{3}}^{3}}{3^{1}}$

Step 7.5.2.1.3.5.5

Evaluate the exponent.

$| x | \leq \frac{\sqrt[4]{14} {\sqrt[4]{3}}^{3}}{3}$

Step 7.5.2.1.4

Simplify the numerator.

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Step 7.5.2.1.4.1

Rewrite ${\sqrt[4]{3}}^{3}$ as $\sqrt[4]{3^{3}}$ .

$| x | \leq \frac{\sqrt[4]{14} \sqrt[4]{3^{3}}}{3}$

Step 7.5.2.1.4.2

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

$| x | \leq \frac{\sqrt[4]{14} \sqrt[4]{27}}{3}$

Step 7.5.2.1.5

Simplify the numerator.

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Step 7.5.2.1.5.1

Combine using the product rule for radicals.

$| x | \leq \frac{\sqrt[4]{14 \cdot 27}}{3}$

Step 7.5.2.1.5.2

Multiply $14$ by $27$ .

$| x | \leq \frac{\sqrt[4]{378}}{3}$

Step 7.6

Write $| x | \leq \frac{\sqrt[4]{378}}{3}$ as a piecewise.

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Step 7.6.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 7.6.2

In the piece where $x$ is non-negative, remove the absolute value.

$x \leq \frac{\sqrt[4]{378}}{3}$

Step 7.6.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 7.6.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x \leq \frac{\sqrt[4]{378}}{3}$

Step 7.6.5

Write as a piecewise.

${\begin{cases} x \leq \frac{\sqrt[4]{378}}{3} & x \geq 0 \\ - x \leq \frac{\sqrt[4]{378}}{3} & x < 0 \end{cases}$

Step 7.7

Find the intersection of $x \leq \frac{\sqrt[4]{378}}{3}$ and $x \geq 0$ .

$0 \leq x \leq \frac{\sqrt[4]{378}}{3}$

Step 7.8

Solve $- x \leq \frac{\sqrt[4]{378}}{3}$ when $x < 0$ .

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Step 7.8.1

Divide each term in $- x \leq \frac{\sqrt[4]{378}}{3}$ by $- 1$ and simplify.

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Step 7.8.1.1

Divide each term in $- x \leq \frac{\sqrt[4]{378}}{3}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{\frac{\sqrt[4]{378}}{3}}{- 1}$

Step 7.8.1.2

Simplify the left side.

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Step 7.8.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{\frac{\sqrt[4]{378}}{3}}{- 1}$

Step 7.8.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{\frac{\sqrt[4]{378}}{3}}{- 1}$

Step 7.8.1.3

Simplify the right side.

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Step 7.8.1.3.1

Move the negative one from the denominator of $\frac{\frac{\sqrt[4]{378}}{3}}{- 1}$ .

$x \geq - 1 \cdot \frac{\sqrt[4]{378}}{3}$

Step 7.8.1.3.2

Rewrite $- 1 \cdot \frac{\sqrt[4]{378}}{3}$ as $- \frac{\sqrt[4]{378}}{3}$ .

$x \geq - \frac{\sqrt[4]{378}}{3}$

Step 7.8.2

Find the intersection of $x \geq - \frac{\sqrt[4]{378}}{3}$ and $x < 0$ .

$- \frac{\sqrt[4]{378}}{3} \leq x < 0$

Step 7.9

Find the union of the solutions.

$- \frac{\sqrt[4]{378}}{3} \leq x \leq \frac{\sqrt[4]{378}}{3}$

Step 8

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[- \frac{\sqrt[4]{378}}{3}, \frac{\sqrt[4]{378}}{3}]$

Set-Builder Notation:

${x | - \frac{\sqrt[4]{378}}{3} \leq x \leq \frac{\sqrt[4]{378}}{3}}$

Step 9