Finite Math Examples

{(1 + x)}^{- \frac{1}{2}}

${(1 + x)}^{- \frac{1}{2}}$

Step 1

Convert expressions with fractional exponents to radicals.

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Step 1.1

Rewrite the expression using the negative exponent rule

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

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

\frac{1}{{(1 + x)}^{\frac{1}{2}}}

$\frac{1}{{(1 + x)}^{\frac{1}{2}}}$

Step 1.2

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{1}{\sqrt{{(1 + x)}^{1}}}

$\frac{1}{\sqrt{{(1 + x)}^{1}}}$

Step 1.3

Anything raised to

1

$1$ is the base itself.

\frac{1}{\sqrt{1 + x}}

$\frac{1}{\sqrt{1 + x}}$

\frac{1}{\sqrt{1 + x}}

$\frac{1}{\sqrt{1 + x}}$

Step 2

Set the denominator in

\frac{1}{\sqrt{1 + x}}

$\frac{1}{\sqrt{1 + x}}$ equal to

0

$0$ to find where the expression is undefined.

\sqrt{1 + x} = 0

$\sqrt{1 + x} = 0$

Step 3

Solve for

x

$x$ .

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Step 3.1

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

{\sqrt{1 + x}}^{2} = 0^{2}

${\sqrt{1 + x}}^{2} = 0^{2}$

Step 3.2

Simplify each side of the equation.

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Step 3.2.1

Use

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

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

\sqrt{1 + x}

$\sqrt{1 + x}$ as

{(1 + x)}^{\frac{1}{2}}

${(1 + x)}^{\frac{1}{2}}$ .

{({(1 + x)}^{\frac{1}{2}})}^{2} = 0^{2}

${({(1 + x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Simplify

{({(1 + x)}^{\frac{1}{2}})}^{2}

${({(1 + x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.2.1.1

Multiply the exponents in

{({(1 + x)}^{\frac{1}{2}})}^{2}

${({(1 + x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.2.1.1.1

Apply the power rule and multiply exponents,

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

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

{(1 + x)}^{\frac{1}{2} \cdot 2} = 0^{2}

${(1 + x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.2.2.1.1.2

Cancel the common factor of

2

$2$ .

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Step 3.2.2.1.1.2.1

Cancel the common factor.

{(1 + x)}^{\frac{1}{2} \cdot 2} = 0^{2}

${(1 + x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.2.2.1.1.2.2

Rewrite the expression.

{(1 + x)}^{1} = 0^{2}

${(1 + x)}^{1} = 0^{2}$

{(1 + x)}^{1} = 0^{2}

${(1 + x)}^{1} = 0^{2}$

{(1 + x)}^{1} = 0^{2}

${(1 + x)}^{1} = 0^{2}$

Step 3.2.2.1.2

Simplify.

1 + x = 0^{2}

$1 + x = 0^{2}$

1 + x = 0^{2}

$1 + x = 0^{2}$

1 + x = 0^{2}

$1 + x = 0^{2}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Raising

0

$0$ to any positive power yields

0

$0$ .

1 + x = 0

$1 + x = 0$

1 + x = 0

$1 + x = 0$

1 + x = 0

$1 + x = 0$

Step 3.3

Subtract

1

$1$ from both sides of the equation.

x = - 1

$x = - 1$

x = - 1

$x = - 1$

Step 4

Set the radicand in

\sqrt{1 + x}

$\sqrt{1 + x}$ less than

0

$0$ to find where the expression is undefined.

1 + x < 0

$1 + x < 0$

Step 5

Subtract

1

$1$ from both sides of the inequality.

x < - 1

$x < - 1$

Step 6

The equation is undefined where the denominator equals

0

$0$ , the argument of a square root is less than

0

$0$ , or the argument of a logarithm is less than or equal to

0

$0$ .

x \leq - 1

$x \leq - 1$

(- \infty, - 1]

$(- \infty, - 1]$

Step 7