Finite Math Examples

$(\frac{5 + \sqrt{5}}{2}, 0)$

Step 1

To find an exponential function, $f (x) = a^{x}$ , containing the point, set $f (x)$ in the function to the $y$ value $0$ of the point, and set $x$ to the $x$ value $\frac{5 + \sqrt{5}}{2}$ of the point.

$0 = a^{\frac{5 + \sqrt{5}}{2}}$

Step 2

Solve the equation for $a$ .

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Step 2.1

Rewrite the equation as $a^{\frac{5 + \sqrt{5}}{2}} = 0$ .

$a^{\frac{5 + \sqrt{5}}{2}} = 0$

Step 2.2

Raise each side of the equation to the power of $\frac{2}{5 + \sqrt{5}}$ to eliminate the fractional exponent on the left side.

${(a^{\frac{5 + \sqrt{5}}{2}})}^{\frac{2}{5 + \sqrt{5}}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3

Simplify the exponent.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Simplify ${(a^{\frac{5 + \sqrt{5}}{2}})}^{\frac{2}{5 + \sqrt{5}}}$ .

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Step 2.3.1.1.1

Multiply $\frac{2}{5 + \sqrt{5}}$ by $\frac{5 - \sqrt{5}}{5 - \sqrt{5}}$ .

${(a^{\frac{5 + \sqrt{5}}{2}})}^{\frac{2}{5 + \sqrt{5}} \cdot \frac{5 - \sqrt{5}}{5 - \sqrt{5}}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.2

Multiply $\frac{2}{5 + \sqrt{5}}$ by $\frac{5 - \sqrt{5}}{5 - \sqrt{5}}$ .

${(a^{\frac{5 + \sqrt{5}}{2}})}^{\frac{2 (5 - \sqrt{5})}{(5 + \sqrt{5}) (5 - \sqrt{5})}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.3

Expand the denominator using the FOIL method.

${(a^{\frac{5 + \sqrt{5}}{2}})}^{\frac{2 (5 - \sqrt{5})}{25 - 5 \sqrt{5} + \sqrt{5} \cdot 5 - {\sqrt{5}}^{2}}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.4

Simplify.

${(a^{\frac{5 + \sqrt{5}}{2}})}^{\frac{2 (5 - \sqrt{5})}{20}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.5

Cancel the common factors.

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Step 2.3.1.1.5.1

Factor $2$ out of $20$ .

${(a^{\frac{5 + \sqrt{5}}{2}})}^{\frac{2 (5 - \sqrt{5})}{2 \cdot 10}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.5.2

Cancel the common factor.

${(a^{\frac{5 + \sqrt{5}}{2}})}^{\frac{2 (5 - \sqrt{5})}{2 \cdot 10}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.5.3

Rewrite the expression.

${(a^{\frac{5 + \sqrt{5}}{2}})}^{\frac{5 - \sqrt{5}}{10}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.6

Multiply the exponents in ${(a^{\frac{5 + \sqrt{5}}{2}})}^{\frac{5 - \sqrt{5}}{10}}$ .

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Step 2.3.1.1.6.1

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

$a^{\frac{5 + \sqrt{5}}{2} \cdot \frac{5 - \sqrt{5}}{10}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.6.2

Multiply $\frac{5 + \sqrt{5}}{2} \cdot \frac{5 - \sqrt{5}}{10}$ .

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Step 2.3.1.1.6.2.1

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

$a^{\frac{(5 + \sqrt{5}) (5 - \sqrt{5})}{2 \cdot 10}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.6.2.2

Multiply $2$ by $10$ .

$a^{\frac{(5 + \sqrt{5}) (5 - \sqrt{5})}{20}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.6.3

Simplify the numerator.

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Step 2.3.1.1.6.3.1

Expand $(5 + \sqrt{5}) (5 - \sqrt{5})$ using the FOIL Method.

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Step 2.3.1.1.6.3.1.1

Apply the distributive property.

$a^{\frac{5 (5 - \sqrt{5}) + \sqrt{5} (5 - \sqrt{5})}{20}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.6.3.1.2

Apply the distributive property.

$a^{\frac{5 \cdot 5 + 5 (- \sqrt{5}) + \sqrt{5} (5 - \sqrt{5})}{20}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.6.3.1.3

Apply the distributive property.

$a^{\frac{5 \cdot 5 + 5 (- \sqrt{5}) + \sqrt{5} \cdot 5 + \sqrt{5} (- \sqrt{5})}{20}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.6.3.2

Simplify and combine like terms.

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Step 2.3.1.1.6.3.2.1

Simplify each term.

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Step 2.3.1.1.6.3.2.1.1

Multiply $5$ by $5$ .

$a^{\frac{25 + 5 (- \sqrt{5}) + \sqrt{5} \cdot 5 + \sqrt{5} (- \sqrt{5})}{20}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.6.3.2.1.2

Multiply $- 1$ by $5$ .

$a^{\frac{25 - 5 \sqrt{5} + \sqrt{5} \cdot 5 + \sqrt{5} (- \sqrt{5})}{20}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.6.3.2.1.3

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

$a^{\frac{25 - 5 \sqrt{5} + 5 \cdot \sqrt{5} + \sqrt{5} (- \sqrt{5})}{20}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.6.3.2.1.4

Multiply $\sqrt{5} (- \sqrt{5})$ .

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Step 2.3.1.1.6.3.2.1.4.1

Raise $\sqrt{5}$ to the power of $1$ .

$a^{\frac{25 - 5 \sqrt{5} + 5 \sqrt{5} - ({\sqrt{5}}^{1} \sqrt{5})}{20}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.6.3.2.1.4.2

Raise $\sqrt{5}$ to the power of $1$ .

$a^{\frac{25 - 5 \sqrt{5} + 5 \sqrt{5} - ({\sqrt{5}}^{1} {\sqrt{5}}^{1})}{20}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.6.3.2.1.4.3

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

$a^{\frac{25 - 5 \sqrt{5} + 5 \sqrt{5} - {\sqrt{5}}^{1 + 1}}{20}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.6.3.2.1.4.4

Add $1$ and $1$ .

$a^{\frac{25 - 5 \sqrt{5} + 5 \sqrt{5} - {\sqrt{5}}^{2}}{20}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.6.3.2.1.5

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

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Step 2.3.1.1.6.3.2.1.5.1

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

$a^{\frac{25 - 5 \sqrt{5} + 5 \sqrt{5} - {(5^{\frac{1}{2}})}^{2}}{20}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.6.3.2.1.5.2

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

$a^{\frac{25 - 5 \sqrt{5} + 5 \sqrt{5} - 5^{\frac{1}{2} \cdot 2}}{20}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.6.3.2.1.5.3

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

$a^{\frac{25 - 5 \sqrt{5} + 5 \sqrt{5} - 5^{\frac{2}{2}}}{20}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.6.3.2.1.5.4

Cancel the common factor of $2$ .

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Step 2.3.1.1.6.3.2.1.5.4.1

Cancel the common factor.

$a^{\frac{25 - 5 \sqrt{5} + 5 \sqrt{5} - 5^{\frac{2}{2}}}{20}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.6.3.2.1.5.4.2

Rewrite the expression.

$a^{\frac{25 - 5 \sqrt{5} + 5 \sqrt{5} - 5^{1}}{20}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.6.3.2.1.5.5

Evaluate the exponent.

$a^{\frac{25 - 5 \sqrt{5} + 5 \sqrt{5} - 1 \cdot 5}{20}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.6.3.2.1.6

Multiply $- 1$ by $5$ .

$a^{\frac{25 - 5 \sqrt{5} + 5 \sqrt{5} - 5}{20}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.6.3.2.2

Subtract $5$ from $25$ .

$a^{\frac{20 - 5 \sqrt{5} + 5 \sqrt{5}}{20}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.6.3.2.3

Add $- 5 \sqrt{5}$ and $5 \sqrt{5}$ .

$a^{\frac{20 + 0}{20}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.6.3.2.4

Add $20$ and $0$ .

$a^{\frac{20}{20}} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.6.4

Divide $20$ by $20$ .

$a^{1} = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.1.1.7

Simplify.

$a = 0^{\frac{2}{5 + \sqrt{5}}}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $0^{\frac{2}{5 + \sqrt{5}}}$ .

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Step 2.3.2.1.1

Multiply $\frac{2}{5 + \sqrt{5}}$ by $\frac{5 - \sqrt{5}}{5 - \sqrt{5}}$ .

$a = 0^{\frac{2}{5 + \sqrt{5}} \cdot \frac{5 - \sqrt{5}}{5 - \sqrt{5}}}$

Step 2.3.2.1.2

Multiply $\frac{2}{5 + \sqrt{5}}$ by $\frac{5 - \sqrt{5}}{5 - \sqrt{5}}$ .

$a = 0^{\frac{2 (5 - \sqrt{5})}{(5 + \sqrt{5}) (5 - \sqrt{5})}}$

Step 2.3.2.1.3

Expand the denominator using the FOIL method.

$a = 0^{\frac{2 (5 - \sqrt{5})}{25 - 5 \sqrt{5} + \sqrt{5} \cdot 5 - {\sqrt{5}}^{2}}}$

Step 2.3.2.1.4

Simplify.

$a = 0^{\frac{2 (5 - \sqrt{5})}{20}}$

Step 2.3.2.1.5

Cancel the common factors.

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Step 2.3.2.1.5.1

Factor $2$ out of $20$ .

$a = 0^{\frac{2 (5 - \sqrt{5})}{2 \cdot 10}}$

Step 2.3.2.1.5.2

Cancel the common factor.

$a = 0^{\frac{2 (5 - \sqrt{5})}{2 \cdot 10}}$

Step 2.3.2.1.5.3

Rewrite the expression.

$a = 0^{\frac{5 - \sqrt{5}}{10}}$

Step 3

Substitute each value for $a$ back into the function $f (x) = a^{x}$ to find each possible exponential function.

$f (x) = {(0^{\frac{5 - \sqrt{5}}{10}})}^{x}$