Linear Algebra Examples

$x^{3.5}$

Step 1

Convert expressions with fractional exponents to radicals.

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Step 1.1

Change $3.5$ into a fraction.

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Step 1.1.1

Multiply by $\frac{10}{10}$ to remove the decimal.

$x^{\frac{10 \cdot 3.5}{10}}$

Step 1.1.2

Multiply $10$ by $3.5$ .

$x^{\frac{35}{10}}$

Step 1.1.3

Cancel the common factor of $35$ and $10$ .

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Step 1.1.3.1

Factor $5$ out of $35$ .

$x^{\frac{5 (7)}{10}}$

Step 1.1.3.2

Cancel the common factors.

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Step 1.1.3.2.1

Factor $5$ out of $10$ .

$x^{\frac{5 \cdot 7}{5 \cdot 2}}$

Step 1.1.3.2.2

Cancel the common factor.

$x^{\frac{5 \cdot 7}{5 \cdot 2}}$

Step 1.1.3.2.3

Rewrite the expression.

$x^{\frac{7}{2}}$

Step 1.2

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

$\sqrt{x^{7}}$

Step 2

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

$x^{7} \geq 0$

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.

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

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.

$x \geq \sqrt[7]{0}$

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 $0^{7}$ .

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

Step 3.2.2.1.2

Pull terms out from under the radical.

$x \geq 0$

Step 4

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

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Step 5