Precalculus Examples

${(4 x - 1)}^{\frac{7}{2}}$

Step 1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\sqrt{{(4 x - 1)}^{7}}$

Step 2

Set the radicand in $\sqrt{{(4 x - 1)}^{7}}$ greater than or equal to $0$ to find where the expression is defined.

${(4 x - 1)}^{7} \geq 0$

Step 3

Solve for $x$ .

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Step 3.1

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

$\sqrt[7]{{(4 x - 1)}^{7}} \geq \sqrt[7]{0}$

Step 3.2

Simplify the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Pull terms out from under the radical.

$4 x - 1 \geq \sqrt[7]{0}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $\sqrt[7]{0}$ .

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Step 3.2.2.1.1

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

$4 x - 1 \geq \sqrt[7]{0^{7}}$

Step 3.2.2.1.2

Pull terms out from under the radical.

$4 x - 1 \geq 0$

Step 3.3

Add $1$ to both sides of the inequality.

$4 x \geq 1$

Step 3.4

Divide each term in $4 x \geq 1$ by $4$ and simplify.

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Step 3.4.1

Divide each term in $4 x \geq 1$ by $4$ .

$\frac{4 x}{4} \geq \frac{1}{4}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $4$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{4 x}{4} \geq \frac{1}{4}$

Step 3.4.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{1}{4}$

Step 4

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

Interval Notation:

$[\frac{1}{4}, \infty)$

Set-Builder Notation:

${x | x \geq \frac{1}{4}}$

Step 5