Algebra Examples

$\frac{x^{2}}{15} - \frac{y^{2}}{5} = 1$

Step 1

Simplify each term in the equation in order to set the right side equal to $1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be $1$ .

$\frac{x^{2}}{15} - \frac{y^{2}}{5} = 1$

Step 2

This is the standard form of a hyperbola. Use this form to determine the eccentricity.

$\frac{{(x - h)}^{2}}{a^{2}} - \frac{{(y - k)}^{2}}{b^{2}} = 1$

Step 3

Match the values in this hyperbola to those of the standard form. The variable $h$ represents the x-offset from the origin, $k$ represents the y-offset from origin, $a$ .

$a = \sqrt{15}$

$b = \sqrt{5}$

$k = 0$

$h = 0$

Step 4

Find the eccentricity by using the following formula.

$\frac{\sqrt{a^{2} + b^{2}}}{a}$

Step 5

Substitute the values of $a$ and $b$ into the formula.

$\frac{\sqrt{{(\sqrt{15})}^{2} + {(\sqrt{5})}^{2}}}{\sqrt{15}}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 6.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$\frac{\sqrt{{(15^{\frac{1}{2}})}^{2} + {\sqrt{5}}^{2}}}{\sqrt{15}}$

Step 6.1.1.2

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

$\frac{\sqrt{15^{\frac{1}{2} \cdot 2} + {\sqrt{5}}^{2}}}{\sqrt{15}}$

Step 6.1.1.3

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

$\frac{\sqrt{15^{\frac{2}{2}} + {\sqrt{5}}^{2}}}{\sqrt{15}}$

Step 6.1.1.4

Cancel the common factor of $2$ .

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Step 6.1.1.4.1

Cancel the common factor.

$\frac{\sqrt{15^{\frac{2}{2}} + {\sqrt{5}}^{2}}}{\sqrt{15}}$

Step 6.1.1.4.2

Rewrite the expression.

$\frac{\sqrt{15^{1} + {\sqrt{5}}^{2}}}{\sqrt{15}}$

Step 6.1.1.5

Evaluate the exponent.

$\frac{\sqrt{15 + {\sqrt{5}}^{2}}}{\sqrt{15}}$

Step 6.1.2

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 6.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{\sqrt{15 + {(5^{\frac{1}{2}})}^{2}}}{\sqrt{15}}$

Step 6.1.2.2

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

$\frac{\sqrt{15 + 5^{\frac{1}{2} \cdot 2}}}{\sqrt{15}}$

Step 6.1.2.3

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

$\frac{\sqrt{15 + 5^{\frac{2}{2}}}}{\sqrt{15}}$

Step 6.1.2.4

Cancel the common factor of $2$ .

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Step 6.1.2.4.1

Cancel the common factor.

$\frac{\sqrt{15 + 5^{\frac{2}{2}}}}{\sqrt{15}}$

Step 6.1.2.4.2

Rewrite the expression.

$\frac{\sqrt{15 + 5^{1}}}{\sqrt{15}}$

Step 6.1.2.5

Evaluate the exponent.

$\frac{\sqrt{15 + 5}}{\sqrt{15}}$

Step 6.1.3

Add $15$ and $5$ .

$\frac{\sqrt{20}}{\sqrt{15}}$

Step 6.1.4

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 6.1.4.1

Factor $4$ out of $20$ .

$\frac{\sqrt{4 (5)}}{\sqrt{15}}$

Step 6.1.4.2

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2} \cdot 5}}{\sqrt{15}}$

Step 6.1.5

Pull terms out from under the radical.

$\frac{2 \sqrt{5}}{\sqrt{15}}$

Step 6.2

Combine $\sqrt{5}$ and $\sqrt{15}$ into a single radical.

$2 \sqrt{\frac{5}{15}}$

Step 6.3

Cancel the common factor of $5$ and $15$ .

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Step 6.3.1

Factor $5$ out of $5$ .

$2 \sqrt{\frac{5 (1)}{15}}$

Step 6.3.2

Cancel the common factors.

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Step 6.3.2.1

Factor $5$ out of $15$ .

$2 \sqrt{\frac{5 \cdot 1}{5 \cdot 3}}$

Step 6.3.2.2

Cancel the common factor.

$2 \sqrt{\frac{5 \cdot 1}{5 \cdot 3}}$

Step 6.3.2.3

Rewrite the expression.

$2 \sqrt{\frac{1}{3}}$

Step 6.4

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$2 \frac{\sqrt{1}}{\sqrt{3}}$

Step 6.5

Any root of $1$ is $1$ .

$2 \frac{1}{\sqrt{3}}$

Step 6.6

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

$2 (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 6.7

Combine and simplify the denominator.

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Step 6.7.1

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

$2 \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 6.7.2

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

$2 \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 6.7.3

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

$2 \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 6.7.4

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

$2 \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 6.7.5

Add $1$ and $1$ .

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

Step 6.7.6

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

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Step 6.7.6.1

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

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

Step 6.7.6.2

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

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

Step 6.7.6.3

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

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

Step 6.7.6.4

Cancel the common factor of $2$ .

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Step 6.7.6.4.1

Cancel the common factor.

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

Step 6.7.6.4.2

Rewrite the expression.

$2 \frac{\sqrt{3}}{3^{1}}$

Step 6.7.6.5

Evaluate the exponent.

$2 \frac{\sqrt{3}}{3}$

Step 6.8

Combine $2$ and $\frac{\sqrt{3}}{3}$ .

$\frac{2 \sqrt{3}}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{3}}{3}$

Decimal Form:

$1.15470053 \dots$

Step 8