Precalculus Examples

$y = \sqrt[7]{5 x}$

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 $x$ .

$0 = \sqrt[7]{5 x}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\sqrt[7]{5 x} = 0$ .

$\sqrt[7]{5 x} = 0$

Step 1.2.2

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $7$ .

${\sqrt[7]{5 x}}^{7} = 0^{7}$

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 $\sqrt[7]{5 x}$ as ${(5 x)}^{\frac{1}{7}}$ .

${({(5 x)}^{\frac{1}{7}})}^{7} = 0^{7}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Simplify ${({(5 x)}^{\frac{1}{7}})}^{7}$ .

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Step 1.2.3.2.1.1

Multiply the exponents in ${({(5 x)}^{\frac{1}{7}})}^{7}$ .

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Step 1.2.3.2.1.1.1

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

${(5 x)}^{\frac{1}{7} \cdot 7} = 0^{7}$

Step 1.2.3.2.1.1.2

Cancel the common factor of $7$ .

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Step 1.2.3.2.1.1.2.1

Cancel the common factor.

${(5 x)}^{\frac{1}{7} \cdot 7} = 0^{7}$

Step 1.2.3.2.1.1.2.2

Rewrite the expression.

${(5 x)}^{1} = 0^{7}$

Step 1.2.3.2.1.2

Simplify.

$5 x = 0^{7}$

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 $0$ .

$5 x = 0$

Step 1.2.4

Divide each term in $5 x = 0$ by $5$ and simplify.

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Step 1.2.4.1

Divide each term in $5 x = 0$ by $5$ .

$\frac{5 x}{5} = \frac{0}{5}$

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

Cancel the common factor of $5$ .

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Step 1.2.4.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{0}{5}$

Step 1.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{5}$

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

Divide $0$ by $5$ .

$x = 0$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0)$

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 $y$ .

$y = \sqrt[7]{5 (0)}$

Step 2.2

Simplify $\sqrt[7]{5 (0)}$ .

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Step 2.2.1

Multiply $5$ by $0$ .

$y = \sqrt[7]{0}$

Step 2.2.2

Rewrite $0$ as $0^{7}$ .

$y = \sqrt[7]{0^{7}}$

Step 2.2.3

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

$y = 0$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 0)$

Step 3

List the intersections.

x-intercept(s): $(0, 0)$

y-intercept(s): $(0, 0)$

Step 4