Linear Algebra Examples

$y x y x y x = 121212$

Step 1

Simplify.

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Step 1.1

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

$x (y^{1} y) x y x = 121212$

Step 1.2

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

$x (y^{1} y^{1}) x y x = 121212$

Step 1.3

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

$x y^{1 + 1} x y x = 121212$

Step 1.4

Add $1$ and $1$ .

$x y^{2} x y x = 121212$

Step 1.5

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

$x^{1} x y^{2} y x = 121212$

Step 1.6

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

$x^{1} x^{1} y^{2} y x = 121212$

Step 1.7

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

$x^{1 + 1} y^{2} y x = 121212$

Step 1.8

Add $1$ and $1$ .

$x^{2} y^{2} y x = 121212$

Step 1.9

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

$x^{2} (y^{1} y^{2}) x = 121212$

Step 1.10

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

$x^{2} y^{1 + 2} x = 121212$

Step 1.11

Add $1$ and $2$ .

$x^{2} y^{3} x = 121212$

Step 1.12

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

$x^{1} x^{2} y^{3} = 121212$

Step 1.13

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

$x^{1 + 2} y^{3} = 121212$

Step 1.14

Add $1$ and $2$ .

$x^{3} y^{3} = 121212$

Step 2

Divide each term in $x^{3} y^{3} = 121212$ by $x^{3}$ and simplify.

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Step 2.1

Divide each term in $x^{3} y^{3} = 121212$ by $x^{3}$ .

$\frac{x^{3} y^{3}}{x^{3}} = \frac{121212}{x^{3}}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $x^{3}$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{x^{3} y^{3}}{x^{3}} = \frac{121212}{x^{3}}$

Step 2.2.1.2

Divide $y^{3}$ by $1$ .

$y^{3} = \frac{121212}{x^{3}}$

Step 3

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

$y = \sqrt[3]{\frac{121212}{x^{3}}}$

Step 4

Simplify $\sqrt[3]{\frac{121212}{x^{3}}}$ .

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Step 4.1

Rewrite $\frac{121212}{x^{3}}$ as ${(\frac{1}{x})}^{3} \cdot 121212$ .

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Step 4.1.1

Factor the perfect power $1^{3}$ out of $121212$ .

$y = \sqrt[3]{\frac{1^{3} \cdot 121212}{x^{3}}}$

Step 4.1.2

Factor the perfect power $x^{3}$ out of $x^{3}$ .

$y = \sqrt[3]{\frac{1^{3} \cdot 121212}{x^{3} \cdot 1}}$

Step 4.1.3

Rearrange the fraction $\frac{1^{3} \cdot 121212}{x^{3} \cdot 1}$ .

$y = \sqrt[3]{{(\frac{1}{x})}^{3} \cdot 121212}$

Step 4.2

Pull terms out from under the radical.

$y = \frac{1}{x} \sqrt[3]{121212}$

Step 4.3

Combine $\frac{1}{x}$ and $\sqrt[3]{121212}$ .

$y = \frac{\sqrt[3]{121212}}{x}$

Step 5

Set the denominator in $\frac{\sqrt[3]{121212}}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 6

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

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Step 7