Linear Algebra Examples

\frac{2}{x^{- 9.5}}

$\frac{2}{x^{- 9.5}}$

Step 1

Convert expressions with fractional exponents to radicals.

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Step 1.1

Change

- 9.5

$- 9.5$ into a fraction.

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Step 1.1.1

Multiply by

\frac{10}{10}

$\frac{10}{10}$ to remove the decimal.

\frac{2}{x^{\frac{10 \cdot - 9.5}{10}}}

$\frac{2}{x^{\frac{10 \cdot - 9.5}{10}}}$

Step 1.1.2

Multiply

10

$10$ by

- 9.5

$- 9.5$ .

\frac{2}{x^{\frac{- 95}{10}}}

$\frac{2}{x^{\frac{- 95}{10}}}$

Step 1.1.3

Move the negative in front of the fraction.

\frac{2}{x^{- \frac{95}{10}}}

$\frac{2}{x^{- \frac{95}{10}}}$

Step 1.1.4

Cancel the common factor of

95

$95$ and

10

$10$ .

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Step 1.1.4.1

Factor

5

$5$ out of

95

$95$ .

\frac{2}{x^{- \frac{5 (19)}{10}}}

$\frac{2}{x^{- \frac{5 (19)}{10}}}$

Step 1.1.4.2

Cancel the common factors.

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Step 1.1.4.2.1

Factor

5

$5$ out of

10

$10$ .

\frac{2}{x^{- \frac{5 \cdot 19}{5 \cdot 2}}}

$\frac{2}{x^{- \frac{5 \cdot 19}{5 \cdot 2}}}$

Step 1.1.4.2.2

Cancel the common factor.

\frac{2}{x^{- \frac{5 \cdot 19}{5 \cdot 2}}}

$\frac{2}{x^{- \frac{5 \cdot 19}{5 \cdot 2}}}$

Step 1.1.4.2.3

Rewrite the expression.

\frac{2}{x^{- \frac{19}{2}}}

$\frac{2}{x^{- \frac{19}{2}}}$

\frac{2}{x^{- \frac{19}{2}}}

$\frac{2}{x^{- \frac{19}{2}}}$

\frac{2}{x^{- \frac{19}{2}}}

$\frac{2}{x^{- \frac{19}{2}}}$

\frac{2}{x^{- \frac{19}{2}}}

$\frac{2}{x^{- \frac{19}{2}}}$

Step 1.2

Rewrite the expression using the negative exponent rule

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

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

\frac{2}{\frac{1}{x^{\frac{19}{2}}}}

$\frac{2}{\frac{1}{x^{\frac{19}{2}}}}$

Step 1.3

Apply the rule

x^{\frac{m}{n}} = \sqrt[n]{x^{m}}

$x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

\frac{2}{\frac{1}{\sqrt{x^{19}}}}

$\frac{2}{\frac{1}{\sqrt{x^{19}}}}$

\frac{2}{\frac{1}{\sqrt{x^{19}}}}

$\frac{2}{\frac{1}{\sqrt{x^{19}}}}$

Step 2

Set the radicand in

\sqrt{x^{19}}

$\sqrt{x^{19}}$ greater than or equal to

0

$0$ to find where the expression is defined.

x^{19} \geq 0

$x^{19} \geq 0$

Step 3

Solve for

x

$x$ .

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Step 3.1

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

\sqrt[19]{x^{19}} \geq \sqrt[19]{0}

$\sqrt[19]{x^{19}} \geq \sqrt[19]{0}$

Step 3.2

Simplify the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Pull terms out from under the radical.

x \geq \sqrt[19]{0}

$x \geq \sqrt[19]{0}$

x \geq \sqrt[19]{0}

$x \geq \sqrt[19]{0}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify

\sqrt[19]{0}

$\sqrt[19]{0}$ .

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Step 3.2.2.1.1

Rewrite

0

$0$ as

0^{19}

$0^{19}$ .

x \geq \sqrt[19]{0^{19}}

$x \geq \sqrt[19]{0^{19}}$

Step 3.2.2.1.2

Pull terms out from under the radical.

x \geq 0

$x \geq 0$

x \geq 0

$x \geq 0$

x \geq 0

$x \geq 0$

x \geq 0

$x \geq 0$

x \geq 0

$x \geq 0$

Step 4

Set the denominator in

\frac{1}{\sqrt{x^{19}}}

$\frac{1}{\sqrt{x^{19}}}$ equal to

0

$0$ to find where the expression is undefined.

\sqrt{x^{19}} = 0

$\sqrt{x^{19}} = 0$

Step 5

Solve for

x

$x$ .

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Step 5.1

To remove the radical on the left side of the equation, square both sides of the equation.

{\sqrt{x^{19}}}^{2} = 0^{2}

${\sqrt{x^{19}}}^{2} = 0^{2}$

Step 5.2

Simplify each side of the equation.

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Step 5.2.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{x^{19}}

$\sqrt{x^{19}}$ as

x^{\frac{19}{2}}

$x^{\frac{19}{2}}$ .

{(x^{\frac{19}{2}})}^{2} = 0^{2}

${(x^{\frac{19}{2}})}^{2} = 0^{2}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Multiply the exponents in

{(x^{\frac{19}{2}})}^{2}

${(x^{\frac{19}{2}})}^{2}$ .

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Step 5.2.2.1.1

Apply the power rule and multiply exponents,

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

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

x^{\frac{19}{2} \cdot 2} = 0^{2}

$x^{\frac{19}{2} \cdot 2} = 0^{2}$

Step 5.2.2.1.2

Cancel the common factor of

2

$2$ .

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Step 5.2.2.1.2.1

Cancel the common factor.

x^{\frac{19}{2} \cdot 2} = 0^{2}

$x^{\frac{19}{2} \cdot 2} = 0^{2}$

Step 5.2.2.1.2.2

Rewrite the expression.

x^{19} = 0^{2}

$x^{19} = 0^{2}$

x^{19} = 0^{2}

$x^{19} = 0^{2}$

x^{19} = 0^{2}

$x^{19} = 0^{2}$

x^{19} = 0^{2}

$x^{19} = 0^{2}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Raising

0

$0$ to any positive power yields

0

$0$ .

x^{19} = 0

$x^{19} = 0$

x^{19} = 0

$x^{19} = 0$

x^{19} = 0

$x^{19} = 0$

Step 5.3

Solve for

x

$x$ .

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Step 5.3.1

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

x = \sqrt[19]{0}

$x = \sqrt[19]{0}$

Step 5.3.2

Simplify

\sqrt[19]{0}

$\sqrt[19]{0}$ .

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Step 5.3.2.1

Rewrite

0

$0$ as

0^{19}

$0^{19}$ .

x = \sqrt[19]{0^{19}}

$x = \sqrt[19]{0^{19}}$

Step 5.3.2.2

Pull terms out from under the radical, assuming real numbers.

x = 0

$x = 0$

x = 0

$x = 0$

x = 0

$x = 0$

x = 0

$x = 0$

Step 6

Set the denominator in

\frac{2}{\frac{1}{\sqrt{x^{19}}}}

$\frac{2}{\frac{1}{\sqrt{x^{19}}}}$ equal to

0

$0$ to find where the expression is undefined.

\frac{1}{\sqrt{x^{19}}} = 0

$\frac{1}{\sqrt{x^{19}}} = 0$

Step 7

Solve for

x

$x$ .

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Step 7.1

Set the numerator equal to zero.

1 = 0

$1 = 0$

Step 7.2

Since

1 \neq 0

$1 \neq 0$ , there are no solutions.

No solution

Step 8

The domain is all values of

x

$x$ that make the expression defined.

Interval Notation:

(0, \infty)

$(0, \infty)$

Set-Builder Notation:

{x | x > 0}

${x | x > 0}$

Step 9