Pre-Algebra Examples

$\sqrt{\frac{e^{- x} (x - 14)}{x + 20}}$

Step 1

Find the $y$ value at $x = 14$ .

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Step 1.1

Replace the variable $x$ with $14$ in the expression.

$f (14) = \frac{\sqrt{e^{- (14)} ((14) - 14) ((14) + 20)}}{(14) + 20}$

Step 1.2

Simplify the result.

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Step 1.2.1

Simplify the numerator.

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Step 1.2.1.1

Multiply $- 1$ by $14$ .

$f (14) = \frac{\sqrt{e^{- 14} (14 - 14) (14 + 20)}}{14 + 20}$

Step 1.2.1.2

Subtract $14$ from $14$ .

$f (14) = \frac{\sqrt{e^{- 14} \cdot (0 (14 + 20))}}{14 + 20}$

Step 1.2.1.3

Combine exponents.

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Step 1.2.1.3.1

Multiply $e^{- 14}$ by $0$ .

$f (14) = \frac{\sqrt{0 (14 + 20)}}{14 + 20}$

Step 1.2.1.3.2

Multiply $0$ by $14 + 20$ .

$f (14) = \frac{\sqrt{0}}{14 + 20}$

Step 1.2.1.4

Rewrite $0$ as $0^{2}$ .

$f (14) = \frac{\sqrt{0^{2}}}{14 + 20}$

Step 1.2.1.5

Pull terms out from under the radical, assuming positive real numbers.

$f (14) = \frac{0}{14 + 20}$

Step 1.2.2

Simplify the expression.

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Step 1.2.2.1

Add $14$ and $20$ .

$f (14) = \frac{0}{34}$

Step 1.2.2.2

Divide $0$ by $34$ .

$f (14) = 0$

Step 1.2.3

The final answer is $0$ .

$0$

Step 1.3

The $y$ value at $x = 14$ is $0$ .

$y = 0$

Step 2

Find the $y$ value at $x = 15$ .

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Step 2.1

Replace the variable $x$ with $15$ in the expression.

$f (15) = \frac{\sqrt{e^{- (15)} ((15) - 14) ((15) + 20)}}{(15) + 20}$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify the numerator.

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Step 2.2.1.1

Multiply $- 1$ by $15$ .

$f (15) = \frac{\sqrt{e^{- 15} (15 - 14) (15 + 20)}}{15 + 20}$

Step 2.2.1.2

Subtract $14$ from $15$ .

$f (15) = \frac{\sqrt{e^{- 15} \cdot (1 (15 + 20))}}{15 + 20}$

Step 2.2.1.3

Multiply $e^{- 15}$ by $1$ .

$f (15) = \frac{\sqrt{e^{- 15} (15 + 20)}}{15 + 20}$

Step 2.2.1.4

Add $15$ and $20$ .

$f (15) = \frac{\sqrt{e^{- 15} \cdot 35}}{15 + 20}$

Step 2.2.1.5

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f (15) = \frac{\sqrt{\frac{1}{e^{15}} \cdot 35}}{15 + 20}$

Step 2.2.1.6

Combine $\frac{1}{e^{15}}$ and $35$ .

$f (15) = \frac{\sqrt{\frac{35}{e^{15}}}}{15 + 20}$

Step 2.2.1.7

Rewrite $\sqrt{\frac{35}{e^{15}}}$ as $\frac{\sqrt{35}}{\sqrt{e^{15}}}$ .

$f (15) = \frac{\frac{\sqrt{35}}{\sqrt{e^{15}}}}{15 + 20}$

Step 2.2.1.8

Simplify the denominator.

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Step 2.2.1.8.1

Rewrite $e^{15}$ as ${(e^{7})}^{2} e$ .

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Step 2.2.1.8.1.1

Factor out $e^{14}$ .

$f (15) = \frac{\frac{\sqrt{35}}{\sqrt{e^{14} e}}}{15 + 20}$

Step 2.2.1.8.1.2

Rewrite $e^{14}$ as ${(e^{7})}^{2}$ .

$f (15) = \frac{\frac{\sqrt{35}}{\sqrt{{(e^{7})}^{2} e}}}{15 + 20}$

Step 2.2.1.8.2

Pull terms out from under the radical.

$f (15) = \frac{\frac{\sqrt{35}}{e^{7} \sqrt{e}}}{15 + 20}$

Step 2.2.1.9

Multiply $\frac{\sqrt{35}}{e^{7} \sqrt{e}}$ by $\frac{\sqrt{e}}{\sqrt{e}}$ .

$f (15) = \frac{\frac{\sqrt{35}}{e^{7} \sqrt{e}} \cdot \frac{\sqrt{e}}{\sqrt{e}}}{15 + 20}$

Step 2.2.1.10

Combine and simplify the denominator.

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Step 2.2.1.10.1

Multiply $\frac{\sqrt{35}}{e^{7} \sqrt{e}}$ by $\frac{\sqrt{e}}{\sqrt{e}}$ .

$f (15) = \frac{\frac{\sqrt{35} \sqrt{e}}{e^{7} \sqrt{e} \sqrt{e}}}{15 + 20}$

Step 2.2.1.10.2

Move $\sqrt{e}$ .

$f (15) = \frac{\frac{\sqrt{35} \sqrt{e}}{e^{7} (\sqrt{e} \sqrt{e})}}{15 + 20}$

Step 2.2.1.10.3

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

$f (15) = \frac{\frac{\sqrt{35} \sqrt{e}}{e^{7} (\sqrt{e} \sqrt{e})}}{15 + 20}$

Step 2.2.1.10.4

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

$f (15) = \frac{\frac{\sqrt{35} \sqrt{e}}{e^{7} (\sqrt{e} \sqrt{e})}}{15 + 20}$

Step 2.2.1.10.5

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

$f (15) = \frac{\frac{\sqrt{35} \sqrt{e}}{e^{7} {\sqrt{e}}^{1 + 1}}}{15 + 20}$

Step 2.2.1.10.6

Add $1$ and $1$ .

$f (15) = \frac{\frac{\sqrt{35} \sqrt{e}}{e^{7} {\sqrt{e}}^{2}}}{15 + 20}$

Step 2.2.1.10.7

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

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Step 2.2.1.10.7.1

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

$f (15) = \frac{\frac{\sqrt{35} \sqrt{e}}{e^{7} {(e^{\frac{1}{2}})}^{2}}}{15 + 20}$

Step 2.2.1.10.7.2

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

$f (15) = \frac{\frac{\sqrt{35} \sqrt{e}}{e^{7} e^{\frac{1}{2} \cdot 2}}}{15 + 20}$

Step 2.2.1.10.7.3

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

$f (15) = \frac{\frac{\sqrt{35} \sqrt{e}}{e^{7} e^{\frac{2}{2}}}}{15 + 20}$

Step 2.2.1.10.7.4

Cancel the common factor of $2$ .

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Step 2.2.1.10.7.4.1

Cancel the common factor.

$f (15) = \frac{\frac{\sqrt{35} \sqrt{e}}{e^{7} e^{\frac{2}{2}}}}{15 + 20}$

Step 2.2.1.10.7.4.2

Rewrite the expression.

$f (15) = \frac{\frac{\sqrt{35} \sqrt{e}}{e^{7} e}}{15 + 20}$

Step 2.2.1.10.7.5

Simplify.

$f (15) = \frac{\frac{\sqrt{35} \sqrt{e}}{e^{7} e}}{15 + 20}$

Step 2.2.1.11

Multiply $e^{7}$ by $e$ by adding the exponents.

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Step 2.2.1.11.1

Multiply $e^{7}$ by $e$ .

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Step 2.2.1.11.1.1

Raise $e$ to the power of $1$ .

$f (15) = \frac{\frac{\sqrt{35} \sqrt{e}}{e^{7} e}}{15 + 20}$

Step 2.2.1.11.1.2

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

$f (15) = \frac{\frac{\sqrt{35} \sqrt{e}}{e^{7 + 1}}}{15 + 20}$

Step 2.2.1.11.2

Add $7$ and $1$ .

$f (15) = \frac{\frac{\sqrt{35} \sqrt{e}}{e^{8}}}{15 + 20}$

Step 2.2.1.12

Combine using the product rule for radicals.

$f (15) = \frac{\frac{\sqrt{35 e}}{e^{8}}}{15 + 20}$

Step 2.2.2

Add $15$ and $20$ .

$f (15) = \frac{\frac{\sqrt{35 e}}{e^{8}}}{35}$

Step 2.2.3

Multiply the numerator by the reciprocal of the denominator.

$f (15) = \frac{\sqrt{35 e}}{e^{8}} \cdot \frac{1}{35}$

Step 2.2.4

Combine.

$f (15) = \frac{\sqrt{35 e} \cdot 1}{e^{8} \cdot 35}$

Step 2.2.5

Simplify the expression.

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Step 2.2.5.1

Multiply $\sqrt{35 e}$ by $1$ .

$f (15) = \frac{\sqrt{35 e}}{e^{8} \cdot 35}$

Step 2.2.5.2

Move $35$ to the left of $e^{8}$ .

$f (15) = \frac{\sqrt{35 e}}{35 e^{8}}$

Step 2.2.6

The final answer is $\frac{\sqrt{35 e}}{35 e^{8}}$ .

$\frac{\sqrt{35 e}}{35 e^{8}}$

Step 2.3

The $y$ value at $x = 15$ is $\frac{\sqrt{35 e}}{35 e^{8}}$ .

$y = \frac{\sqrt{35 e}}{35 e^{8}}$

Step 3

Find the $y$ value at $x = 16$ .

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Step 3.1

Replace the variable $x$ with $16$ in the expression.

$f (16) = \frac{\sqrt{e^{- (16)} ((16) - 14) ((16) + 20)}}{(16) + 20}$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify the numerator.

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Step 3.2.1.1

Multiply $- 1$ by $16$ .

$f (16) = \frac{\sqrt{e^{- 16} (16 - 14) (16 + 20)}}{16 + 20}$

Step 3.2.1.2

Subtract $14$ from $16$ .

$f (16) = \frac{\sqrt{e^{- 16} \cdot (2 (16 + 20))}}{16 + 20}$

Step 3.2.1.3

Add $16$ and $20$ .

$f (16) = \frac{\sqrt{e^{- 16} \cdot 2 \cdot 36}}{16 + 20}$

Step 3.2.1.4

Multiply $36$ by $2$ .

$f (16) = \frac{\sqrt{e^{- 16} \cdot 72}}{16 + 20}$

Step 3.2.1.5

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f (16) = \frac{\sqrt{\frac{1}{e^{16}} \cdot 72}}{16 + 20}$

Step 3.2.1.6

Combine $\frac{1}{e^{16}}$ and $72$ .

$f (16) = \frac{\sqrt{\frac{72}{e^{16}}}}{16 + 20}$

Step 3.2.1.7

Rewrite $\sqrt{\frac{72}{e^{16}}}$ as $\frac{\sqrt{72}}{\sqrt{e^{16}}}$ .

$f (16) = \frac{\frac{\sqrt{72}}{\sqrt{e^{16}}}}{16 + 20}$

Step 3.2.1.8

Simplify the numerator.

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Step 3.2.1.8.1

Rewrite $72$ as $6^{2} \cdot 2$ .

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Step 3.2.1.8.1.1

Factor $36$ out of $72$ .

$f (16) = \frac{\frac{\sqrt{36 (2)}}{\sqrt{e^{16}}}}{16 + 20}$

Step 3.2.1.8.1.2

Rewrite $36$ as $6^{2}$ .

$f (16) = \frac{\frac{\sqrt{6^{2} \cdot 2}}{\sqrt{e^{16}}}}{16 + 20}$

Step 3.2.1.8.2

Pull terms out from under the radical.

$f (16) = \frac{\frac{6 \sqrt{2}}{\sqrt{e^{16}}}}{16 + 20}$

Step 3.2.1.9

Simplify the denominator.

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Step 3.2.1.9.1

Rewrite $e^{16}$ as ${(e^{8})}^{2}$ .

$f (16) = \frac{\frac{6 \sqrt{2}}{\sqrt{{(e^{8})}^{2}}}}{16 + 20}$

Step 3.2.1.9.2

Pull terms out from under the radical, assuming positive real numbers.

$f (16) = \frac{\frac{6 \sqrt{2}}{e^{8}}}{16 + 20}$

Step 3.2.2

Add $16$ and $20$ .

$f (16) = \frac{\frac{6 \sqrt{2}}{e^{8}}}{36}$

Step 3.2.3

Multiply the numerator by the reciprocal of the denominator.

$f (16) = \frac{6 \sqrt{2}}{e^{8}} \cdot \frac{1}{36}$

Step 3.2.4

Combine.

$f (16) = \frac{6 \sqrt{2} \cdot 1}{e^{8} \cdot 36}$

Step 3.2.5

Cancel the common factor of $6$ and $36$ .

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Step 3.2.5.1

Factor $6$ out of $6 \sqrt{2} \cdot 1$ .

$f (16) = \frac{6 (\sqrt{2} \cdot 1)}{e^{8} \cdot 36}$

Step 3.2.5.2

Cancel the common factors.

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Step 3.2.5.2.1

Factor $6$ out of $e^{8} \cdot 36$ .

$f (16) = \frac{6 (\sqrt{2} \cdot 1)}{6 (e^{8} \cdot 6)}$

Step 3.2.5.2.2

Cancel the common factor.

$f (16) = \frac{6 (\sqrt{2} \cdot 1)}{6 (e^{8} \cdot 6)}$

Step 3.2.5.2.3

Rewrite the expression.

$f (16) = \frac{\sqrt{2} \cdot 1}{e^{8} \cdot 6}$

Step 3.2.6

Simplify the expression.

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Step 3.2.6.1

Multiply $\sqrt{2}$ by $1$ .

$f (16) = \frac{\sqrt{2}}{e^{8} \cdot 6}$

Step 3.2.6.2

Move $6$ to the left of $e^{8}$ .

$f (16) = \frac{\sqrt{2}}{6 e^{8}}$

Step 3.2.7

The final answer is $\frac{\sqrt{2}}{6 e^{8}}$ .

$\frac{\sqrt{2}}{6 e^{8}}$

Step 3.3

The $y$ value at $x = 16$ is $\frac{\sqrt{2}}{6 e^{8}}$ .

$y = \frac{\sqrt{2}}{6 e^{8}}$

Step 4

List the points to graph.

$(14, 0), (15, 0.00009348), (16, 0.00007906)$

Step 5

Select a few points to graph.

$\begin{array}{c} x & y \\ 14 & 0 \\ 15 & 0 \\ 16 & 0 \end{array}$

Step 6