Find the Domain ( square root of 7x^2)÷( square root of 3x)

$\sqrt{7 x^{2}} \div \sqrt{3 x}$

Step 1

Set the radicand in $\sqrt{7 x^{2}}$ greater than or equal to $0$ to find where the expression is defined.

$7 x^{2} \geq 0$

Step 2

Solve for $x$ .

Divide each term in $7 x^{2} \geq 0$ by $7$ and simplify.

Divide each term in $7 x^{2} \geq 0$ by $7$ .

$\frac{7 x^{2}}{7} \geq \frac{0}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x^{2}}{7} \geq \frac{0}{7}$

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

$x^{2} \geq \frac{0}{7}$

Simplify the right side.

Divide $0$ by $7$ .

$x^{2} \geq 0$

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

All real numbers

Step 3

Set the radicand in $\sqrt{3 x}$ greater than or equal to $0$ to find where the expression is defined.

$3 x \geq 0$

Step 4

Divide each term in $3 x \geq 0$ by $3$ and simplify.

Divide each term in $3 x \geq 0$ by $3$ .

$\frac{3 x}{3} \geq \frac{0}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{0}{3}$

Divide $x$ by $1$ .

$x \geq \frac{0}{3}$

Simplify the right side.

Divide $0$ by $3$ .

$x \geq 0$

Step 5

Set the denominator in $\frac{\sqrt{7 x^{2}}}{\sqrt{3 x}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{3 x} = 0$

Step 6

Solve for $x$ .

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

${\sqrt{3 x}}^{2} = 0^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x}$ as ${(3 x)}^{\frac{1}{2}}$ .

${({(3 x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Simplify the left side.

Simplify ${({(3 x)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(3 x)}^{\frac{1}{2}})}^{2}$ .

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

${(3 x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(3 x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Rewrite the expression.

${(3 x)}^{1} = 0^{2}$

Simplify.

$3 x = 0^{2}$

Simplify the right side.

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

$3 x = 0$

Divide each term in $3 x = 0$ by $3$ and simplify.

Divide each term in $3 x = 0$ by $3$ .

$\frac{3 x}{3} = \frac{0}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{0}{3}$

Divide $x$ by $1$ .

$x = \frac{0}{3}$

Simplify the right side.

Divide $0$ by $3$ .

$x = 0$

Step 7

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

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${x | x > 0}$

Step 8

Find the Domain ( square root of 7x^2)÷( square root of 3x)

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