Precalculus Examples

$f (x) = \sqrt[3]{9 + \sqrt{1 + x}}$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \sqrt[3]{9 + \sqrt{1 + x}}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\sqrt[3]{9 + \sqrt{1 + x}} = 0$ .

$\sqrt[3]{9 + \sqrt{1 + x}} = 0$

Step 1.2.2

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

${\sqrt[3]{9 + \sqrt{1 + x}}}^{3} = 0^{3}$

Step 1.2.3

Simplify each side of the equation.

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Step 1.2.3.1

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

${({(9 + \sqrt{1 + x})}^{\frac{1}{3}})}^{3} = 0^{3}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

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

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Step 1.2.3.2.1.1

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

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Step 1.2.3.2.1.1.1

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

${(9 + \sqrt{1 + x})}^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 1.2.3.2.1.1.2

Cancel the common factor of $3$ .

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Step 1.2.3.2.1.1.2.1

Cancel the common factor.

${(9 + \sqrt{1 + x})}^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 1.2.3.2.1.1.2.2

Rewrite the expression.

${(9 + \sqrt{1 + x})}^{1} = 0^{3}$

Step 1.2.3.2.1.2

Simplify.

$9 + \sqrt{1 + x} = 0^{3}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

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

$9 + \sqrt{1 + x} = 0$

Step 1.2.4

Subtract $9$ from both sides of the equation.

$\sqrt{1 + x} = - 9$

Step 1.2.5

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

${\sqrt{1 + x}}^{2} = {(- 9)}^{2}$

Step 1.2.6

Simplify each side of the equation.

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Step 1.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 + x}$ as ${(1 + x)}^{\frac{1}{2}}$ .

${({(1 + x)}^{\frac{1}{2}})}^{2} = {(- 9)}^{2}$

Step 1.2.6.2

Simplify the left side.

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Step 1.2.6.2.1

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

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Step 1.2.6.2.1.1

Multiply the exponents in ${({(1 + x)}^{\frac{1}{2}})}^{2}$ .

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Step 1.2.6.2.1.1.1

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

${(1 + x)}^{\frac{1}{2} \cdot 2} = {(- 9)}^{2}$

Step 1.2.6.2.1.1.2

Cancel the common factor of $2$ .

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Step 1.2.6.2.1.1.2.1

Cancel the common factor.

${(1 + x)}^{\frac{1}{2} \cdot 2} = {(- 9)}^{2}$

Step 1.2.6.2.1.1.2.2

Rewrite the expression.

${(1 + x)}^{1} = {(- 9)}^{2}$

Step 1.2.6.2.1.2

Simplify.

$1 + x = {(- 9)}^{2}$

Step 1.2.6.3

Simplify the right side.

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Step 1.2.6.3.1

Raise $- 9$ to the power of $2$ .

$1 + x = 81$

Step 1.2.7

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.7.1

Subtract $1$ from both sides of the equation.

$x = 81 - 1$

Step 1.2.7.2

Subtract $1$ from $81$ .

$x = 80$

Step 1.2.8

Exclude the solutions that do not make $0 = \sqrt[3]{9 + \sqrt{1 + x}}$ true.

$No solution$

Step 1.3

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

x-intercept(s): $None$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \sqrt[3]{9 + \sqrt{1 + 0}}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \sqrt[3]{9 + \sqrt{1 + 0}}$

Step 2.2.2

Simplify $\sqrt[3]{9 + \sqrt{1 + 0}}$ .

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Step 2.2.2.1

Simplify by adding zeros.

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Step 2.2.2.1.1

Add $1$ and $0$ .

$y = \sqrt[3]{9 + \sqrt{1}}$

Step 2.2.2.1.2

Any root of $1$ is $1$ .

$y = \sqrt[3]{9 + 1}$

Step 2.2.2.2

Add $9$ and $1$ .

$y = \sqrt[3]{10}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \sqrt[3]{10})$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, \sqrt[3]{10})$

Step 4