Linear Algebra Examples

\frac{{(x y)}^{- 3}}{{(x^{- 5} y)}^{3}}

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

Step 1

Set the denominator in

\frac{{(x y)}^{- 3}}{{(x^{- 5} y)}^{3}}

$\frac{{(x y)}^{- 3}}{{(x^{- 5} y)}^{3}}$ equal to

0

$0$ to find where the expression is undefined.

{(x^{- 5} y)}^{3} = 0

${(x^{- 5} y)}^{3} = 0$

Step 2

Solve for

x

$x$ .

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Step 2.1

Simplify

{(x^{- 5} y)}^{3}

${(x^{- 5} y)}^{3}$ .

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Step 2.1.1

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

{(\frac{1}{x^{5}} y)}^{3} = 0

${(\frac{1}{x^{5}} y)}^{3} = 0$

Step 2.1.2

Combine

\frac{1}{x^{5}}

$\frac{1}{x^{5}}$ and

y

$y$ .

{(\frac{y}{x^{5}})}^{3} = 0

${(\frac{y}{x^{5}})}^{3} = 0$

Step 2.1.3

Apply basic rules of exponents.

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Step 2.1.3.1

Apply the product rule to

\frac{y}{x^{5}}

$\frac{y}{x^{5}}$ .

\frac{y^{3}}{{(x^{5})}^{3}} = 0

$\frac{y^{3}}{{(x^{5})}^{3}} = 0$

Step 2.1.3.2

Multiply the exponents in

{(x^{5})}^{3}

${(x^{5})}^{3}$ .

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Step 2.1.3.2.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

\frac{y^{3}}{x^{5 \cdot 3}} = 0

$\frac{y^{3}}{x^{5 \cdot 3}} = 0$

Step 2.1.3.2.2

Multiply

5

$5$ by

3

$3$ .

\frac{y^{3}}{x^{15}} = 0

$\frac{y^{3}}{x^{15}} = 0$

\frac{y^{3}}{x^{15}} = 0

$\frac{y^{3}}{x^{15}} = 0$

\frac{y^{3}}{x^{15}} = 0

$\frac{y^{3}}{x^{15}} = 0$

\frac{y^{3}}{x^{15}} = 0

$\frac{y^{3}}{x^{15}} = 0$

Step 2.2

Multiply both sides of the equation by

x^{15}

$x^{15}$ .

y^{3} = x^{15} (0)

$y^{3} = x^{15} (0)$

Step 2.3

Rewrite the equation as

x^{15} (0) = y^{3}

$x^{15} (0) = y^{3}$ .

x^{15} (0) = y^{3}

$x^{15} (0) = y^{3}$

Step 2.4

Multiply

x^{15}

$x^{15}$ by

0

$0$ .

0 = y^{3}

$0 = y^{3}$

Step 2.5

The variable

x

$x$ got canceled.

All real numbers

All real numbers

Step 3

Set the base in

{(x y)}^{- 3}

${(x y)}^{- 3}$ equal to

0

$0$ to find where the expression is undefined.

x y = 0

$x y = 0$

Step 4

Divide each term in

x y = 0

$x y = 0$ by

y

$y$ and simplify.

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Step 4.1

Divide each term in

x y = 0

$x y = 0$ by

y

$y$ .

\frac{x y}{y} = \frac{0}{y}

$\frac{x y}{y} = \frac{0}{y}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of

y

$y$ .

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Step 4.2.1.1

Cancel the common factor.

\frac{x y}{y} = \frac{0}{y}

$\frac{x y}{y} = \frac{0}{y}$

Step 4.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{0}{y}

$x = \frac{0}{y}$

x = \frac{0}{y}

$x = \frac{0}{y}$

x = \frac{0}{y}

$x = \frac{0}{y}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide

0

$0$ by

y

$y$ .

x = 0

$x = 0$

x = 0

$x = 0$

x = 0

$x = 0$

Step 5

The domain is all values of

x

$x$ that make the expression defined.

Interval Notation:

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

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

Set-Builder Notation:

{x | x \neq 0}

${x | x \neq 0}$

[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$