Find the Domain ( square root of 15(x-1))÷( square root of 2x^2)

$\sqrt{15 (x - 1)} \div \sqrt{2 x^{2}}$

Set the radicand in $\sqrt{15 (x - 1)}$ greater than or equal to $0$ to find where the expression is defined.

$15 (x - 1) \geq 0$

Solve for $x$ .

Divide each term in $15 (x - 1) \geq 0$ by $15$ and simplify.

Divide each term in $15 (x - 1) \geq 0$ by $15$ .

$\frac{15 (x - 1)}{15} \geq \frac{0}{15}$

Simplify the left side.

Cancel the common factor of $15$ .

Cancel the common factor.

$\frac{15 (x - 1)}{15} \geq \frac{0}{15}$

Divide $x - 1$ by $1$ .

$x - 1 \geq \frac{0}{15}$

Simplify the right side.

Divide $0$ by $15$ .

$x - 1 \geq 0$

Add $1$ to both sides of the inequality.

$x \geq 1$

Set the radicand in $\sqrt{2 x^{2}}$ greater than or equal to $0$ to find where the expression is defined.

$2 x^{2} \geq 0$

Solve for $x$ .

Divide each term in $2 x^{2} \geq 0$ by $2$ and simplify.

Divide each term in $2 x^{2} \geq 0$ by $2$ .

$\frac{2 x^{2}}{2} \geq \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x^{2}}{2} \geq \frac{0}{2}$

Divide $x^{2}$ by $1$ .

$x^{2} \geq \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$x^{2} \geq 0$

Since the left side has an even power, it is always positive for all real numbers.

All real numbers

Set the denominator in $\frac{\sqrt{15 (x - 1)}}{\sqrt{2 x^{2}}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{2 x^{2}} = 0$

Solve for $x$ .

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

${\sqrt{2 x^{2}}}^{2} = 0^{2}$

Simplify each side of the equation.

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

${({(2 x^{2})}^{\frac{1}{2}})}^{2} = 0^{2}$

Simplify the left side.

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

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

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

${(2 x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(2 x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Rewrite the expression.

${(2 x^{2})}^{1} = 0^{2}$

Simplify.

$2 x^{2} = 0^{2}$

Simplify the right side.

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

$2 x^{2} = 0$

Solve for $x$ .

Divide each term in $2 x^{2} = 0$ by $2$ and simplify.

Divide each term in $2 x^{2} = 0$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{0}{2}$

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$x^{2} = 0$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

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

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

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

Interval Notation:

$[1, \infty)$

Set-Builder Notation:

${x | x \geq 1}$

Find the Domain ( square root of 15(x-1))÷( square root of 2x^2)

Related Questions

Support LCMGCF.com by disabling your adblocker.