Linear Algebra Examples

$\frac{3 \sqrt{27 x^{27}}}{3 x^{- 2}} \cdot \frac{{(- 3 x)}^{3}}{3 x^{- 1}}$

Step 1

Set the radicand in $\sqrt{27 x^{27}}$ greater than or equal to $0$ to find where the expression is defined.

$27 x^{27} \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Divide each term in $27 x^{27} \geq 0$ by $27$ and simplify.

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Step 2.1.1

Divide each term in $27 x^{27} \geq 0$ by $27$ .

$\frac{27 x^{27}}{27} \geq \frac{0}{27}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $27$ .

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{27 x^{27}}{27} \geq \frac{0}{27}$

Step 2.1.2.1.2

Divide $x^{27}$ by $1$ .

$x^{27} \geq \frac{0}{27}$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Divide $0$ by $27$ .

$x^{27} \geq 0$

Step 2.2

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

$\sqrt[27]{x^{27}} \geq \sqrt[27]{0}$

Step 2.3

Simplify the equation.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Pull terms out from under the radical.

$x \geq \sqrt[27]{0}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $\sqrt[27]{0}$ .

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Step 2.3.2.1.1

Rewrite $0$ as $0^{27}$ .

$x \geq \sqrt[27]{0^{27}}$

Step 2.3.2.1.2

Pull terms out from under the radical.

$x \geq 0$

Step 3

Set the denominator in $\frac{3 \sqrt{27 x^{27}}}{3 x^{- 2}}$ equal to $0$ to find where the expression is undefined.

$3 x^{- 2} = 0$

Step 4

Solve for $x$ .

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Step 4.1

Simplify $3 x^{- 2}$ .

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Step 4.1.1

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

$3 \frac{1}{x^{2}} = 0$

Step 4.1.2

Combine $3$ and $\frac{1}{x^{2}}$ .

$\frac{3}{x^{2}} = 0$

Step 4.2

Set the numerator equal to zero.

$3 = 0$

Step 4.3

Since $3 \neq 0$ , there are no solutions.

No solution

Step 5

Set the denominator in $\frac{{(- 3 x)}^{3}}{3 x^{- 1}}$ equal to $0$ to find where the expression is undefined.

$3 x^{- 1} = 0$

Step 6

Solve for $x$ .

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Step 6.1

Simplify $3 x^{- 1}$ .

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Step 6.1.1

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

$3 \frac{1}{x} = 0$

Step 6.1.2

Combine $3$ and $\frac{1}{x}$ .

$\frac{3}{x} = 0$

Step 6.2

Set the numerator equal to zero.

$3 = 0$

Step 6.3

Since $3 \neq 0$ , there are no solutions.

No solution

Step 7

Set the base in $x^{- 2}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 8

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

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${x | x > 0}$

Step 9