Pre-Algebra Examples

$5 x^{2} + 8 y^{2} = 36$

Step 1

Find the standard form of the ellipse.

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Step 1.1

Divide each term by $36$ to make the right side equal to one.

$\frac{5 x^{2}}{36} + \frac{8 y^{2}}{36} = \frac{36}{36}$

Step 1.2

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 $1$ .

$\frac{x^{2}}{\frac{36}{5}} + \frac{y^{2}}{\frac{9}{2}} = 1$

Step 2

This is the form of an ellipse. Use this form to determine the values used to find the center along with the major and minor axis of the ellipse.

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

Step 3

Match the values in this ellipse to those of the standard form. The variable $a$ represents the radius of the major axis of the ellipse, $b$ represents the radius of the minor axis of the ellipse, $h$ represents the x-offset from the origin, and $k$ represents the y-offset from the origin.

$a = \frac{6 \sqrt{5}}{5}$

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

$k = 0$

$h = 0$

Step 4

The center of an ellipse follows the form of $(h, k)$ . Substitute in the values of $h$ and $k$ .

$(0, 0)$

Step 5

Find $c$ , the distance from the center to a focus.

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Step 5.1

Find the distance from the center to a focus of the ellipse by using the following formula.

$\sqrt{a^{2} - b^{2}}$

Step 5.2

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

$\sqrt{{(\frac{6 \sqrt{5}}{5})}^{2} - {(\frac{3 \sqrt{2}}{2})}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.3.1.1

Apply the product rule to $\frac{6 \sqrt{5}}{5}$ .

$\sqrt{\frac{{(6 \sqrt{5})}^{2}}{5^{2}} - {(\frac{3 \sqrt{2}}{2})}^{2}}$

Step 5.3.1.2

Apply the product rule to $6 \sqrt{5}$ .

$\sqrt{\frac{6^{2} {\sqrt{5}}^{2}}{5^{2}} - {(\frac{3 \sqrt{2}}{2})}^{2}}$

Step 5.3.2

Simplify the numerator.

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Step 5.3.2.1

Raise $6$ to the power of $2$ .

$\sqrt{\frac{36 {\sqrt{5}}^{2}}{5^{2}} - {(\frac{3 \sqrt{2}}{2})}^{2}}$

Step 5.3.2.2

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

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Step 5.3.2.2.1

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

$\sqrt{\frac{36 {(5^{\frac{1}{2}})}^{2}}{5^{2}} - {(\frac{3 \sqrt{2}}{2})}^{2}}$

Step 5.3.2.2.2

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

$\sqrt{\frac{36 \cdot 5^{\frac{1}{2} \cdot 2}}{5^{2}} - {(\frac{3 \sqrt{2}}{2})}^{2}}$

Step 5.3.2.2.3

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

$\sqrt{\frac{36 \cdot 5^{\frac{2}{2}}}{5^{2}} - {(\frac{3 \sqrt{2}}{2})}^{2}}$

Step 5.3.2.2.4

Cancel the common factor of $2$ .

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Step 5.3.2.2.4.1

Cancel the common factor.

$\sqrt{\frac{36 \cdot 5^{\frac{2}{2}}}{5^{2}} - {(\frac{3 \sqrt{2}}{2})}^{2}}$

Step 5.3.2.2.4.2

Rewrite the expression.

$\sqrt{\frac{36 \cdot 5^{1}}{5^{2}} - {(\frac{3 \sqrt{2}}{2})}^{2}}$

Step 5.3.2.2.5

Evaluate the exponent.

$\sqrt{\frac{36 \cdot 5}{5^{2}} - {(\frac{3 \sqrt{2}}{2})}^{2}}$

Step 5.3.3

Reduce the expression by cancelling the common factors.

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Step 5.3.3.1

Raise $5$ to the power of $2$ .

$\sqrt{\frac{36 \cdot 5}{25} - {(\frac{3 \sqrt{2}}{2})}^{2}}$

Step 5.3.3.2

Multiply $36$ by $5$ .

$\sqrt{\frac{180}{25} - {(\frac{3 \sqrt{2}}{2})}^{2}}$

Step 5.3.3.3

Cancel the common factor of $180$ and $25$ .

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Step 5.3.3.3.1

Factor $5$ out of $180$ .

$\sqrt{\frac{5 (36)}{25} - {(\frac{3 \sqrt{2}}{2})}^{2}}$

Step 5.3.3.3.2

Cancel the common factors.

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Step 5.3.3.3.2.1

Factor $5$ out of $25$ .

$\sqrt{\frac{5 \cdot 36}{5 \cdot 5} - {(\frac{3 \sqrt{2}}{2})}^{2}}$

Step 5.3.3.3.2.2

Cancel the common factor.

$\sqrt{\frac{5 \cdot 36}{5 \cdot 5} - {(\frac{3 \sqrt{2}}{2})}^{2}}$

Step 5.3.3.3.2.3

Rewrite the expression.

$\sqrt{\frac{36}{5} - {(\frac{3 \sqrt{2}}{2})}^{2}}$

Step 5.3.4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.3.4.1

Apply the product rule to $\frac{3 \sqrt{2}}{2}$ .

$\sqrt{\frac{36}{5} - \frac{{(3 \sqrt{2})}^{2}}{2^{2}}}$

Step 5.3.4.2

Apply the product rule to $3 \sqrt{2}$ .

$\sqrt{\frac{36}{5} - \frac{3^{2} {\sqrt{2}}^{2}}{2^{2}}}$

Step 5.3.5

Simplify the numerator.

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Step 5.3.5.1

Raise $3$ to the power of $2$ .

$\sqrt{\frac{36}{5} - \frac{9 {\sqrt{2}}^{2}}{2^{2}}}$

Step 5.3.5.2

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

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Step 5.3.5.2.1

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

$\sqrt{\frac{36}{5} - \frac{9 {(2^{\frac{1}{2}})}^{2}}{2^{2}}}$

Step 5.3.5.2.2

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

$\sqrt{\frac{36}{5} - \frac{9 \cdot 2^{\frac{1}{2} \cdot 2}}{2^{2}}}$

Step 5.3.5.2.3

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

$\sqrt{\frac{36}{5} - \frac{9 \cdot 2^{\frac{2}{2}}}{2^{2}}}$

Step 5.3.5.2.4

Cancel the common factor of $2$ .

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Step 5.3.5.2.4.1

Cancel the common factor.

$\sqrt{\frac{36}{5} - \frac{9 \cdot 2^{\frac{2}{2}}}{2^{2}}}$

Step 5.3.5.2.4.2

Rewrite the expression.

$\sqrt{\frac{36}{5} - \frac{9 \cdot 2^{1}}{2^{2}}}$

Step 5.3.5.2.5

Evaluate the exponent.

$\sqrt{\frac{36}{5} - \frac{9 \cdot 2}{2^{2}}}$

Step 5.3.6

Reduce the expression by cancelling the common factors.

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Step 5.3.6.1

Raise $2$ to the power of $2$ .

$\sqrt{\frac{36}{5} - \frac{9 \cdot 2}{4}}$

Step 5.3.6.2

Multiply $9$ by $2$ .

$\sqrt{\frac{36}{5} - \frac{18}{4}}$

Step 5.3.6.3

Cancel the common factor of $18$ and $4$ .

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Step 5.3.6.3.1

Factor $2$ out of $18$ .

$\sqrt{\frac{36}{5} - \frac{2 (9)}{4}}$

Step 5.3.6.3.2

Cancel the common factors.

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Step 5.3.6.3.2.1

Factor $2$ out of $4$ .

$\sqrt{\frac{36}{5} - \frac{2 \cdot 9}{2 \cdot 2}}$

Step 5.3.6.3.2.2

Cancel the common factor.

$\sqrt{\frac{36}{5} - \frac{2 \cdot 9}{2 \cdot 2}}$

Step 5.3.6.3.2.3

Rewrite the expression.

$\sqrt{\frac{36}{5} - \frac{9}{2}}$

Step 5.3.7

To write $\frac{36}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{\frac{36}{5} \cdot \frac{2}{2} - \frac{9}{2}}$

Step 5.3.8

To write $- \frac{9}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\sqrt{\frac{36}{5} \cdot \frac{2}{2} - \frac{9}{2} \cdot \frac{5}{5}}$

Step 5.3.9

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.9.1

Multiply $\frac{36}{5}$ by $\frac{2}{2}$ .

$\sqrt{\frac{36 \cdot 2}{5 \cdot 2} - \frac{9}{2} \cdot \frac{5}{5}}$

Step 5.3.9.2

Multiply $5$ by $2$ .

$\sqrt{\frac{36 \cdot 2}{10} - \frac{9}{2} \cdot \frac{5}{5}}$

Step 5.3.9.3

Multiply $\frac{9}{2}$ by $\frac{5}{5}$ .

$\sqrt{\frac{36 \cdot 2}{10} - \frac{9 \cdot 5}{2 \cdot 5}}$

Step 5.3.9.4

Multiply $2$ by $5$ .

$\sqrt{\frac{36 \cdot 2}{10} - \frac{9 \cdot 5}{10}}$

Step 5.3.10

Combine the numerators over the common denominator.

$\sqrt{\frac{36 \cdot 2 - 9 \cdot 5}{10}}$

Step 5.3.11

Simplify the numerator.

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Step 5.3.11.1

Multiply $36$ by $2$ .

$\sqrt{\frac{72 - 9 \cdot 5}{10}}$

Step 5.3.11.2

Multiply $- 9$ by $5$ .

$\sqrt{\frac{72 - 45}{10}}$

Step 5.3.11.3

Subtract $45$ from $72$ .

$\sqrt{\frac{27}{10}}$

Step 5.3.12

Rewrite $\sqrt{\frac{27}{10}}$ as $\frac{\sqrt{27}}{\sqrt{10}}$ .

$\frac{\sqrt{27}}{\sqrt{10}}$

Step 5.3.13

Simplify the numerator.

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Step 5.3.13.1

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 5.3.13.1.1

Factor $9$ out of $27$ .

$\frac{\sqrt{9 (3)}}{\sqrt{10}}$

Step 5.3.13.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2} \cdot 3}}{\sqrt{10}}$

Step 5.3.13.2

Pull terms out from under the radical.

$\frac{3 \sqrt{3}}{\sqrt{10}}$

Step 5.3.14

Multiply $\frac{3 \sqrt{3}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{3 \sqrt{3}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$

Step 5.3.15

Combine and simplify the denominator.

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Step 5.3.15.1

Multiply $\frac{3 \sqrt{3}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{3 \sqrt{3} \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 5.3.15.2

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{3 \sqrt{3} \sqrt{10}}{{\sqrt{10}}^{1} \sqrt{10}}$

Step 5.3.15.3

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{3 \sqrt{3} \sqrt{10}}{{\sqrt{10}}^{1} {\sqrt{10}}^{1}}$

Step 5.3.15.4

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

$\frac{3 \sqrt{3} \sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Step 5.3.15.5

Add $1$ and $1$ .

$\frac{3 \sqrt{3} \sqrt{10}}{{\sqrt{10}}^{2}}$

Step 5.3.15.6

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 5.3.15.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\frac{3 \sqrt{3} \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

Step 5.3.15.6.2

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

$\frac{3 \sqrt{3} \sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

Step 5.3.15.6.3

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

$\frac{3 \sqrt{3} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 5.3.15.6.4

Cancel the common factor of $2$ .

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Step 5.3.15.6.4.1

Cancel the common factor.

$\frac{3 \sqrt{3} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 5.3.15.6.4.2

Rewrite the expression.

$\frac{3 \sqrt{3} \sqrt{10}}{10^{1}}$

Step 5.3.15.6.5

Evaluate the exponent.

$\frac{3 \sqrt{3} \sqrt{10}}{10}$

Step 5.3.16

Simplify the numerator.

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Step 5.3.16.1

Combine using the product rule for radicals.

$\frac{3 \sqrt{10 \cdot 3}}{10}$

Step 5.3.16.2

Multiply $10$ by $3$ .

$\frac{3 \sqrt{30}}{10}$

Step 6

Find the vertices.

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Step 6.1

The first vertex of an ellipse can be found by adding $a$ to $h$ .

$(h + a, k)$

Step 6.2

Substitute the known values of $h$ , $a$ , and $k$ into the formula.

$(0 + \frac{6 \sqrt{5}}{5}, 0)$

Step 6.3

Simplify.

$(\frac{6 \sqrt{5}}{5}, 0)$

Step 6.4

The second vertex of an ellipse can be found by subtracting $a$ from $h$ .

$(h - a, k)$

Step 6.5

Substitute the known values of $h$ , $a$ , and $k$ into the formula.

$(0 - (\frac{6 \sqrt{5}}{5}), 0)$

Step 6.6

Simplify.

$(- \frac{6 \sqrt{5}}{5}, 0)$

Step 6.7

Ellipses have two vertices.

${Vertex}_{1}$ : $(\frac{6 \sqrt{5}}{5}, 0)$

${Vertex}_{2}$ : $(- \frac{6 \sqrt{5}}{5}, 0)$

${Vertex}_{1}$ : $(\frac{6 \sqrt{5}}{5}, 0)$

${Vertex}_{2}$ : $(- \frac{6 \sqrt{5}}{5}, 0)$

Step 7

Find the foci.

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Step 7.1

The first focus of an ellipse can be found by adding $c$ to $h$ .

$(h + c, k)$

Step 7.2

Substitute the known values of $h$ , $c$ , and $k$ into the formula.

$(0 + \frac{3 \sqrt{30}}{10}, 0)$

Step 7.3

Simplify.

$(\frac{3 \sqrt{30}}{10}, 0)$

Step 7.4

The second focus of an ellipse can be found by subtracting $c$ from $h$ .

$(h - c, k)$

Step 7.5

Substitute the known values of $h$ , $c$ , and $k$ into the formula.

$(0 - (\frac{3 \sqrt{30}}{10}), 0)$

Step 7.6

Simplify.

$(- \frac{3 \sqrt{30}}{10}, 0)$

Step 7.7

Ellipses have two foci.

${Focus}_{1}$ : $(\frac{3 \sqrt{30}}{10}, 0)$

${Focus}_{2}$ : $(- \frac{3 \sqrt{30}}{10}, 0)$

${Focus}_{1}$ : $(\frac{3 \sqrt{30}}{10}, 0)$

${Focus}_{2}$ : $(- \frac{3 \sqrt{30}}{10}, 0)$

Step 8

Find the eccentricity.

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Step 8.1

Find the eccentricity by using the following formula.

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

Step 8.2

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

$\frac{\sqrt{{(\frac{6 \sqrt{5}}{5})}^{2} - {(\frac{3 \sqrt{2}}{2})}^{2}}}{\frac{6 \sqrt{5}}{5}}$

Step 8.3

Simplify.

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Step 8.3.1

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{{(\frac{6 \sqrt{5}}{5})}^{2} - {(\frac{3 \sqrt{2}}{2})}^{2}} \frac{5}{6 \sqrt{5}}$

Step 8.3.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 8.3.2.1

Apply the product rule to $\frac{6 \sqrt{5}}{5}$ .

$\sqrt{\frac{{(6 \sqrt{5})}^{2}}{5^{2}} - {(\frac{3 \sqrt{2}}{2})}^{2}} \frac{5}{6 \sqrt{5}}$

Step 8.3.2.2

Apply the product rule to $6 \sqrt{5}$ .

$\sqrt{\frac{6^{2} {\sqrt{5}}^{2}}{5^{2}} - {(\frac{3 \sqrt{2}}{2})}^{2}} \frac{5}{6 \sqrt{5}}$

Step 8.3.3

Simplify the numerator.

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Step 8.3.3.1

Raise $6$ to the power of $2$ .

$\sqrt{\frac{36 {\sqrt{5}}^{2}}{5^{2}} - {(\frac{3 \sqrt{2}}{2})}^{2}} \frac{5}{6 \sqrt{5}}$

Step 8.3.3.2

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

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Step 8.3.3.2.1

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

$\sqrt{\frac{36 {(5^{\frac{1}{2}})}^{2}}{5^{2}} - {(\frac{3 \sqrt{2}}{2})}^{2}} \frac{5}{6 \sqrt{5}}$

Step 8.3.3.2.2

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

$\sqrt{\frac{36 \cdot 5^{\frac{1}{2} \cdot 2}}{5^{2}} - {(\frac{3 \sqrt{2}}{2})}^{2}} \frac{5}{6 \sqrt{5}}$

Step 8.3.3.2.3

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

$\sqrt{\frac{36 \cdot 5^{\frac{2}{2}}}{5^{2}} - {(\frac{3 \sqrt{2}}{2})}^{2}} \frac{5}{6 \sqrt{5}}$

Step 8.3.3.2.4

Cancel the common factor of $2$ .

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Step 8.3.3.2.4.1

Cancel the common factor.

$\sqrt{\frac{36 \cdot 5^{\frac{2}{2}}}{5^{2}} - {(\frac{3 \sqrt{2}}{2})}^{2}} \frac{5}{6 \sqrt{5}}$

Step 8.3.3.2.4.2

Rewrite the expression.

$\sqrt{\frac{36 \cdot 5^{1}}{5^{2}} - {(\frac{3 \sqrt{2}}{2})}^{2}} \frac{5}{6 \sqrt{5}}$

Step 8.3.3.2.5

Evaluate the exponent.

$\sqrt{\frac{36 \cdot 5}{5^{2}} - {(\frac{3 \sqrt{2}}{2})}^{2}} \frac{5}{6 \sqrt{5}}$

Step 8.3.4

Reduce the expression by cancelling the common factors.

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Step 8.3.4.1

Raise $5$ to the power of $2$ .

$\sqrt{\frac{36 \cdot 5}{25} - {(\frac{3 \sqrt{2}}{2})}^{2}} \frac{5}{6 \sqrt{5}}$

Step 8.3.4.2

Multiply $36$ by $5$ .

$\sqrt{\frac{180}{25} - {(\frac{3 \sqrt{2}}{2})}^{2}} \frac{5}{6 \sqrt{5}}$

Step 8.3.4.3

Cancel the common factor of $180$ and $25$ .

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Step 8.3.4.3.1

Factor $5$ out of $180$ .

$\sqrt{\frac{5 (36)}{25} - {(\frac{3 \sqrt{2}}{2})}^{2}} \frac{5}{6 \sqrt{5}}$

Step 8.3.4.3.2

Cancel the common factors.

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Step 8.3.4.3.2.1

Factor $5$ out of $25$ .

$\sqrt{\frac{5 \cdot 36}{5 \cdot 5} - {(\frac{3 \sqrt{2}}{2})}^{2}} \frac{5}{6 \sqrt{5}}$

Step 8.3.4.3.2.2

Cancel the common factor.

$\sqrt{\frac{5 \cdot 36}{5 \cdot 5} - {(\frac{3 \sqrt{2}}{2})}^{2}} \frac{5}{6 \sqrt{5}}$

Step 8.3.4.3.2.3

Rewrite the expression.

$\sqrt{\frac{36}{5} - {(\frac{3 \sqrt{2}}{2})}^{2}} \frac{5}{6 \sqrt{5}}$

Step 8.3.5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 8.3.5.1

Apply the product rule to $\frac{3 \sqrt{2}}{2}$ .

$\sqrt{\frac{36}{5} - \frac{{(3 \sqrt{2})}^{2}}{2^{2}}} \frac{5}{6 \sqrt{5}}$

Step 8.3.5.2

Apply the product rule to $3 \sqrt{2}$ .

$\sqrt{\frac{36}{5} - \frac{3^{2} {\sqrt{2}}^{2}}{2^{2}}} \frac{5}{6 \sqrt{5}}$

Step 8.3.6

Simplify the numerator.

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Step 8.3.6.1

Raise $3$ to the power of $2$ .

$\sqrt{\frac{36}{5} - \frac{9 {\sqrt{2}}^{2}}{2^{2}}} \frac{5}{6 \sqrt{5}}$

Step 8.3.6.2

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

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Step 8.3.6.2.1

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

$\sqrt{\frac{36}{5} - \frac{9 {(2^{\frac{1}{2}})}^{2}}{2^{2}}} \frac{5}{6 \sqrt{5}}$

Step 8.3.6.2.2

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

$\sqrt{\frac{36}{5} - \frac{9 \cdot 2^{\frac{1}{2} \cdot 2}}{2^{2}}} \frac{5}{6 \sqrt{5}}$

Step 8.3.6.2.3

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

$\sqrt{\frac{36}{5} - \frac{9 \cdot 2^{\frac{2}{2}}}{2^{2}}} \frac{5}{6 \sqrt{5}}$

Step 8.3.6.2.4

Cancel the common factor of $2$ .

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Step 8.3.6.2.4.1

Cancel the common factor.

$\sqrt{\frac{36}{5} - \frac{9 \cdot 2^{\frac{2}{2}}}{2^{2}}} \frac{5}{6 \sqrt{5}}$

Step 8.3.6.2.4.2

Rewrite the expression.

$\sqrt{\frac{36}{5} - \frac{9 \cdot 2^{1}}{2^{2}}} \frac{5}{6 \sqrt{5}}$

Step 8.3.6.2.5

Evaluate the exponent.

$\sqrt{\frac{36}{5} - \frac{9 \cdot 2}{2^{2}}} \frac{5}{6 \sqrt{5}}$

Step 8.3.7

Reduce the expression by cancelling the common factors.

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Step 8.3.7.1

Raise $2$ to the power of $2$ .

$\sqrt{\frac{36}{5} - \frac{9 \cdot 2}{4}} \frac{5}{6 \sqrt{5}}$

Step 8.3.7.2

Multiply $9$ by $2$ .

$\sqrt{\frac{36}{5} - \frac{18}{4}} \frac{5}{6 \sqrt{5}}$

Step 8.3.7.3

Cancel the common factor of $18$ and $4$ .

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Step 8.3.7.3.1

Factor $2$ out of $18$ .

$\sqrt{\frac{36}{5} - \frac{2 (9)}{4}} \frac{5}{6 \sqrt{5}}$

Step 8.3.7.3.2

Cancel the common factors.

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Step 8.3.7.3.2.1

Factor $2$ out of $4$ .

$\sqrt{\frac{36}{5} - \frac{2 \cdot 9}{2 \cdot 2}} \frac{5}{6 \sqrt{5}}$

Step 8.3.7.3.2.2

Cancel the common factor.

$\sqrt{\frac{36}{5} - \frac{2 \cdot 9}{2 \cdot 2}} \frac{5}{6 \sqrt{5}}$

Step 8.3.7.3.2.3

Rewrite the expression.

$\sqrt{\frac{36}{5} - \frac{9}{2}} \frac{5}{6 \sqrt{5}}$

Step 8.3.8

To write $\frac{36}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{\frac{36}{5} \cdot \frac{2}{2} - \frac{9}{2}} \frac{5}{6 \sqrt{5}}$

Step 8.3.9

To write $- \frac{9}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\sqrt{\frac{36}{5} \cdot \frac{2}{2} - \frac{9}{2} \cdot \frac{5}{5}} \frac{5}{6 \sqrt{5}}$

Step 8.3.10

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.3.10.1

Multiply $\frac{36}{5}$ by $\frac{2}{2}$ .

$\sqrt{\frac{36 \cdot 2}{5 \cdot 2} - \frac{9}{2} \cdot \frac{5}{5}} \frac{5}{6 \sqrt{5}}$

Step 8.3.10.2

Multiply $5$ by $2$ .

$\sqrt{\frac{36 \cdot 2}{10} - \frac{9}{2} \cdot \frac{5}{5}} \frac{5}{6 \sqrt{5}}$

Step 8.3.10.3

Multiply $\frac{9}{2}$ by $\frac{5}{5}$ .

$\sqrt{\frac{36 \cdot 2}{10} - \frac{9 \cdot 5}{2 \cdot 5}} \frac{5}{6 \sqrt{5}}$

Step 8.3.10.4

Multiply $2$ by $5$ .

$\sqrt{\frac{36 \cdot 2}{10} - \frac{9 \cdot 5}{10}} \frac{5}{6 \sqrt{5}}$

Step 8.3.11

Combine the numerators over the common denominator.

$\sqrt{\frac{36 \cdot 2 - 9 \cdot 5}{10}} \frac{5}{6 \sqrt{5}}$

Step 8.3.12

Simplify the numerator.

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Step 8.3.12.1

Multiply $36$ by $2$ .

$\sqrt{\frac{72 - 9 \cdot 5}{10}} \frac{5}{6 \sqrt{5}}$

Step 8.3.12.2

Multiply $- 9$ by $5$ .

$\sqrt{\frac{72 - 45}{10}} \frac{5}{6 \sqrt{5}}$

Step 8.3.12.3

Subtract $45$ from $72$ .

$\sqrt{\frac{27}{10}} \frac{5}{6 \sqrt{5}}$

Step 8.3.13

Rewrite $\sqrt{\frac{27}{10}}$ as $\frac{\sqrt{27}}{\sqrt{10}}$ .

$\frac{\sqrt{27}}{\sqrt{10}} \cdot \frac{5}{6 \sqrt{5}}$

Step 8.3.14

Simplify the numerator.

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Step 8.3.14.1

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 8.3.14.1.1

Factor $9$ out of $27$ .

$\frac{\sqrt{9 (3)}}{\sqrt{10}} \cdot \frac{5}{6 \sqrt{5}}$

Step 8.3.14.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2} \cdot 3}}{\sqrt{10}} \cdot \frac{5}{6 \sqrt{5}}$

Step 8.3.14.2

Pull terms out from under the radical.

$\frac{3 \sqrt{3}}{\sqrt{10}} \cdot \frac{5}{6 \sqrt{5}}$

Step 8.3.15

Simplify terms.

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Step 8.3.15.1

Cancel the common factor of $3$ .

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Step 8.3.15.1.1

Factor $3$ out of $3 \sqrt{3}$ .

$\frac{3 (\sqrt{3})}{\sqrt{10}} \cdot \frac{5}{6 \sqrt{5}}$

Step 8.3.15.1.2

Factor $3$ out of $6 \sqrt{5}$ .

$\frac{3 (\sqrt{3})}{\sqrt{10}} \cdot \frac{5}{3 (2 \sqrt{5})}$

Step 8.3.15.1.3

Cancel the common factor.

$\frac{3 \sqrt{3}}{\sqrt{10}} \cdot \frac{5}{3 (2 \sqrt{5})}$

Step 8.3.15.1.4

Rewrite the expression.

$\frac{\sqrt{3}}{\sqrt{10}} \cdot \frac{5}{2 \sqrt{5}}$

Step 8.3.15.2

Multiply $\frac{\sqrt{3}}{\sqrt{10}}$ by $\frac{5}{2 \sqrt{5}}$ .

$\frac{\sqrt{3} \cdot 5}{\sqrt{10} (2 \sqrt{5})}$

Step 8.3.15.3

Combine using the product rule for radicals.

$\frac{\sqrt{3} \cdot 5}{2 \sqrt{10 \cdot 5}}$

Step 8.3.15.4

Simplify the expression.

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Step 8.3.15.4.1

Multiply $10$ by $5$ .

$\frac{\sqrt{3} \cdot 5}{2 \sqrt{50}}$

Step 8.3.15.4.2

Move $5$ to the left of $\sqrt{3}$ .

$\frac{5 \sqrt{3}}{2 \sqrt{50}}$

Step 8.3.16

Simplify the denominator.

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Step 8.3.16.1

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 8.3.16.1.1

Factor $25$ out of $50$ .

$\frac{5 \sqrt{3}}{2 \sqrt{25 (2)}}$

Step 8.3.16.1.2

Rewrite $25$ as $5^{2}$ .

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

Step 8.3.16.2

Pull terms out from under the radical.

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

Step 8.3.16.3

Multiply $2$ by $5$ .

$\frac{5 \sqrt{3}}{10 \sqrt{2}}$

Step 8.3.17

Cancel the common factor of $5$ and $10$ .

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Step 8.3.17.1

Factor $5$ out of $5 \sqrt{3}$ .

$\frac{5 (\sqrt{3})}{10 \sqrt{2}}$

Step 8.3.17.2

Cancel the common factors.

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Step 8.3.17.2.1

Factor $5$ out of $10 \sqrt{2}$ .

$\frac{5 (\sqrt{3})}{5 (2 \sqrt{2})}$

Step 8.3.17.2.2

Cancel the common factor.

$\frac{5 \sqrt{3}}{5 (2 \sqrt{2})}$

Step 8.3.17.2.3

Rewrite the expression.

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

Step 8.3.18

Multiply $\frac{\sqrt{3}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 8.3.19

Combine and simplify the denominator.

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Step 8.3.19.1

Multiply $\frac{\sqrt{3}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 8.3.19.2

Move $\sqrt{2}$ .

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

Step 8.3.19.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 8.3.19.4

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 8.3.19.5

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

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

Step 8.3.19.6

Add $1$ and $1$ .

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

Step 8.3.19.7

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

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Step 8.3.19.7.1

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

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

Step 8.3.19.7.2

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

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

Step 8.3.19.7.3

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

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

Step 8.3.19.7.4

Cancel the common factor of $2$ .

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Step 8.3.19.7.4.1

Cancel the common factor.

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

Step 8.3.19.7.4.2

Rewrite the expression.

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

Step 8.3.19.7.5

Evaluate the exponent.

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

Step 8.3.20

Simplify the numerator.

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Step 8.3.20.1

Combine using the product rule for radicals.

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

Step 8.3.20.2

Multiply $3$ by $2$ .

$\frac{\sqrt{6}}{2 \cdot 2}$

Step 8.3.21

Multiply $2$ by $2$ .

$\frac{\sqrt{6}}{4}$

Step 9

These values represent the important values for graphing and analyzing an ellipse.

Center: $(0, 0)$

${Vertex}_{1}$ : $(\frac{6 \sqrt{5}}{5}, 0)$

${Vertex}_{2}$ : $(- \frac{6 \sqrt{5}}{5}, 0)$

${Focus}_{1}$ : $(\frac{3 \sqrt{30}}{10}, 0)$

${Focus}_{2}$ : $(- \frac{3 \sqrt{30}}{10}, 0)$

Eccentricity: $\frac{\sqrt{6}}{4}$

Step 10