Precalculus Examples

$\sqrt{\sqrt[3]{9 - x}}$

Step 1

Set the radicand in $\sqrt{\sqrt[3]{9 - x}}$ greater than or equal to $0$ to find where the expression is defined.

$\sqrt[3]{9 - x} \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

To remove the radical on the left side of the inequality, cube both sides of the inequality.

${\sqrt[3]{9 - x}}^{3} \geq 0^{3}$

Step 2.2

Simplify each side of the inequality.

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{9 - x}$ as ${(9 - x)}^{\frac{1}{3}}$ .

${({(9 - x)}^{\frac{1}{3}})}^{3} \geq 0^{3}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify ${({(9 - x)}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.2.1.1

Multiply the exponents in ${({(9 - x)}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.2.1.1.1

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

${(9 - x)}^{\frac{1}{3} \cdot 3} \geq 0^{3}$

Step 2.2.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.2.2.1.1.2.1

Cancel the common factor.

${(9 - x)}^{\frac{1}{3} \cdot 3} \geq 0^{3}$

Step 2.2.2.1.1.2.2

Rewrite the expression.

${(9 - x)}^{1} \geq 0^{3}$

Step 2.2.2.1.2

Simplify.

$9 - x \geq 0^{3}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Raising $0$ to any positive power yields $0$ .

$9 - x \geq 0$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Subtract $9$ from both sides of the inequality.

$- x \geq - 9$

Step 2.3.2

Divide each term in $- x \geq - 9$ by $- 1$ and simplify.

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Step 2.3.2.1

Divide each term in $- x \geq - 9$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 9}{- 1}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 9}{- 1}$

Step 2.3.2.2.2

Divide $x$ by $1$ .

$x \leq \frac{- 9}{- 1}$

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Divide $- 9$ by $- 1$ .

$x \leq 9$

Step 3

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

Interval Notation:

$(- \infty, 9]$

Set-Builder Notation:

${x | x \leq 9}$

Step 4