Algebra Examples

$\sqrt{5 x^{9} + 5}$

Step 1

Find the domain for $\sqrt{5 x^{9} + 5}$ .

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Step 1.1

Set the radicand in $\sqrt{5 x^{9} + 5}$ greater than or equal to $0$ to find where the expression is defined.

$5 x^{9} + 5 \geq 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Subtract $5$ from both sides of the inequality.

$5 x^{9} \geq - 5$

Step 1.2.2

Divide each term in $5 x^{9} \geq - 5$ by $5$ and simplify.

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Step 1.2.2.1

Divide each term in $5 x^{9} \geq - 5$ by $5$ .

$\frac{5 x^{9}}{5} \geq \frac{- 5}{5}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $5$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{5 x^{9}}{5} \geq \frac{- 5}{5}$

Step 1.2.2.2.1.2

Divide $x^{9}$ by $1$ .

$x^{9} \geq \frac{- 5}{5}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $- 5$ by $5$ .

$x^{9} \geq - 1$

Step 1.2.3

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

$\sqrt[9]{x^{9}} \geq \sqrt[9]{- 1}$

Step 1.2.4

Simplify the equation.

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Step 1.2.4.1

Simplify the left side.

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Step 1.2.4.1.1

Pull terms out from under the radical.

$x \geq \sqrt[9]{- 1}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify $\sqrt[9]{- 1}$ .

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Step 1.2.4.2.1.1

Rewrite $- 1$ as ${(- 1)}^{9}$ .

$x \geq \sqrt[9]{{(- 1)}^{9}}$

Step 1.2.4.2.1.2

Pull terms out from under the radical.

$x \geq - 1$

Step 1.3

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

Interval Notation:

$[- 1, \infty)$

Set-Builder Notation:

${x | x \geq - 1}$

Interval Notation:

$[- 1, \infty)$

Set-Builder Notation:

${x | x \geq - 1}$

Step 2

To find the square root end point, substitute the $x$ value $- 1$ , which is the terminal value in the domain, into $\sqrt{5 x^{9} + 5}$ .

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Step 2.1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = \sqrt{5 {(- 1)}^{9} + 5}$

Step 2.2

Simplify the result.

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Step 2.2.1

Raise $- 1$ to the power of $9$ .

$f (- 1) = \sqrt{5 \cdot - 1 + 5}$

Step 2.2.2

Multiply $5$ by $- 1$ .

$f (- 1) = \sqrt{- 5 + 5}$

Step 2.2.3

Add $- 5$ and $5$ .

$f (- 1) = \sqrt{0}$

Step 2.2.4

Rewrite $0$ as $0^{2}$ .

$f (- 1) = \sqrt{0^{2}}$

Step 2.2.5

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

$f (- 1) = 0$

Step 2.2.6

The final answer is $0$ .

$y = 0$

Step 3

The square root end point is $(- 1, 0)$ .

$(- 1, 0)$

Step 4