Linear Algebra Examples

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

Step 1

Set the denominator in $\frac{{(x y)}^{- 3}}{{(x^{- 5} y)}^{3}}$ equal to $0$ to find where the expression is undefined.

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

Step 2

Solve for $x$ .

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Step 2.1

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

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Step 2.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.1.2

Combine $\frac{1}{x^{5}}$ and $y$ .

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

Step 2.1.3

Apply basic rules of exponents.

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Step 2.1.3.1

Apply the product rule to $\frac{y}{x^{5}}$ .

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

Step 2.1.3.2

Multiply the exponents in ${(x^{5})}^{3}$ .

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Step 2.1.3.2.1

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

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

Step 2.1.3.2.2

Multiply $5$ by $3$ .

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

Step 2.2

Multiply both sides of the equation by $x^{15}$ .

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

Step 2.3

Rewrite the equation as $x^{15} (0) = y^{3}$ .

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

Step 2.4

Multiply $x^{15}$ by $0$ .

$0 = y^{3}$

Step 2.5

The variable $x$ got canceled.

All real numbers

Step 3

Set the base in ${(x y)}^{- 3}$ equal to $0$ to find where the expression is undefined.

$x y = 0$

Step 4

Divide each term in $x y = 0$ by $y$ and simplify.

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Step 4.1

Divide each term in $x y = 0$ by $y$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $y$ .

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Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Divide $x$ by $1$ .

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

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $y$ .

$x = 0$

Step 5

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

Interval Notation:

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

Set-Builder Notation:

${x | x \neq 0}$