Algebra Examples

$\sqrt{x + 1} \div \sqrt{3 - x}$

Step 1

Write $\sqrt{x + 1} \div \sqrt{3 - x}$ as an equation.

$y = \sqrt{x + 1} \div \sqrt{3 - x}$

Step 2

Find the x-intercepts.

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Step 2.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \sqrt{x + 1} \div \sqrt{3 - x}$

Step 2.2

Solve the equation.

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Step 2.2.1

Set the numerator equal to zero.

$\sqrt{x + 1} = 0$

Step 2.2.2

Solve the equation for $x$ .

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Step 2.2.2.1

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

${\sqrt{x + 1}}^{2} = 0^{2}$

Step 2.2.2.2

Simplify each side of the equation.

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Step 2.2.2.2.1

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

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

Step 2.2.2.2.2

Simplify the left side.

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Step 2.2.2.2.2.1

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

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Step 2.2.2.2.2.1.1

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

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Step 2.2.2.2.2.1.1.1

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

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

Step 2.2.2.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.2.2.2.1.1.2.1

Cancel the common factor.

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

Step 2.2.2.2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.2.2.2.1.2

Simplify.

$x + 1 = 0^{2}$

Step 2.2.2.2.3

Simplify the right side.

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Step 2.2.2.2.3.1

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

$x + 1 = 0$

Step 2.2.2.3

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(- 1, 0)$

Step 3

Find the y-intercepts.

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Step 3.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \sqrt{(0) + 1} \div \sqrt{3 - (0)}$

Step 3.2

Simplify $\sqrt{(0) + 1} \div \sqrt{3 - (0)}$ .

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Step 3.2.1

Rewrite the division as a fraction.

$y = \frac{\sqrt{(0) + 1}}{\sqrt{3 - (0)}}$

Step 3.2.2

Simplify the numerator.

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Step 3.2.2.1

Add $0$ and $1$ .

$y = \frac{\sqrt{1}}{\sqrt{3 - (0)}}$

Step 3.2.2.2

Any root of $1$ is $1$ .

$y = \frac{1}{\sqrt{3 - (0)}}$

Step 3.2.3

Simplify the denominator.

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Step 3.2.3.1

Multiply $- 1$ by $0$ .

$y = \frac{1}{\sqrt{3 + 0}}$

Step 3.2.3.2

Add $3$ and $0$ .

$y = \frac{1}{\sqrt{3}}$

Step 3.2.4

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 3.2.5

Combine and simplify the denominator.

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Step 3.2.5.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 3.2.5.2

Raise $\sqrt{3}$ to the power of $1$ .

$y = \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 3.2.5.3

Raise $\sqrt{3}$ to the power of $1$ .

$y = \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 3.2.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 3.2.5.5

Add $1$ and $1$ .

$y = \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 3.2.5.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.2.5.6.1

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

$y = \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 3.2.5.6.2

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

$y = \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 3.2.5.6.3

Combine $\frac{1}{2}$ and $2$ .

$y = \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.2.5.6.4

Cancel the common factor of $2$ .

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Step 3.2.5.6.4.1

Cancel the common factor.

$y = \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.2.5.6.4.2

Rewrite the expression.

$y = \frac{\sqrt{3}}{3^{1}}$

Step 3.2.5.6.5

Evaluate the exponent.

$y = \frac{\sqrt{3}}{3}$

Step 3.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{\sqrt{3}}{3})$

Step 4

List the intersections.

x-intercept(s): $(- 1, 0)$

y-intercept(s): $(0, \frac{\sqrt{3}}{3})$

Step 5