Calculus Examples

$\frac{d y}{d x} = - 8 x^{7} e^{- x^{8}}$ , $y (0) = 8$

Step 1

Rewrite the equation.

$d y = - 8 x^{7} e^{- x^{8}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int - 8 x^{7} e^{- x^{8}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int - 8 x^{7} e^{- x^{8}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $- 8$ is constant with respect to $x$ , move $- 8$ out of the integral.

$y + C_{1} = - 8 \int x^{7} e^{- x^{8}} d x$

Step 2.3.2

Let $u_{1} = x^{2}$ . Then $d u_{1} = 2 x d x$ , so $\frac{1}{2} d u_{1} = x d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 2.3.2.1

Let $u_{1} = x^{2}$ . Find $\frac{d u_{1}}{d x}$ .

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Step 2.3.2.1.1

Differentiate $x^{2}$ .

$\frac{d}{d x} [x^{2}]$

Step 2.3.2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x$

Step 2.3.2.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$y + C_{1} = - 8 \int {\sqrt{u_{1}}}^{6} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Rewrite ${\sqrt{u_{1}}}^{6}$ as ${u_{1}}^{3}$ .

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Step 2.3.3.1.1

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

$y + C_{1} = - 8 \int {({u_{1}}^{\frac{1}{2}})}^{6} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 2.3.3.1.2

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

$y + C_{1} = - 8 \int {u_{1}}^{\frac{1}{2} \cdot 6} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 2.3.3.1.3

Combine $\frac{1}{2}$ and $6$ .

$y + C_{1} = - 8 \int {u_{1}}^{\frac{6}{2}} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 2.3.3.1.4

Cancel the common factor of $6$ and $2$ .

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Step 2.3.3.1.4.1

Factor $2$ out of $6$ .

$y + C_{1} = - 8 \int {u_{1}}^{\frac{2 \cdot 3}{2}} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 2.3.3.1.4.2

Cancel the common factors.

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Step 2.3.3.1.4.2.1

Factor $2$ out of $2$ .

$y + C_{1} = - 8 \int {u_{1}}^{\frac{2 \cdot 3}{2 (1)}} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 2.3.3.1.4.2.2

Cancel the common factor.

$y + C_{1} = - 8 \int {u_{1}}^{\frac{2 \cdot 3}{2 \cdot 1}} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 2.3.3.1.4.2.3

Rewrite the expression.

$y + C_{1} = - 8 \int {u_{1}}^{\frac{3}{1}} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 2.3.3.1.4.2.4

Divide $3$ by $1$ .

$y + C_{1} = - 8 \int {u_{1}}^{3} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 2.3.3.2

Rewrite ${\sqrt{u_{1}}}^{8}$ as ${u_{1}}^{4}$ .

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Step 2.3.3.2.1

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

$y + C_{1} = - 8 \int {u_{1}}^{3} e^{- {({u_{1}}^{\frac{1}{2}})}^{8}} \frac{1}{2} d u_{1}$

Step 2.3.3.2.2

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

$y + C_{1} = - 8 \int {u_{1}}^{3} e^{- {u_{1}}^{\frac{1}{2} \cdot 8}} \frac{1}{2} d u_{1}$

Step 2.3.3.2.3

Combine $\frac{1}{2}$ and $8$ .

$y + C_{1} = - 8 \int {u_{1}}^{3} e^{- {u_{1}}^{\frac{8}{2}}} \frac{1}{2} d u_{1}$

Step 2.3.3.2.4

Cancel the common factor of $8$ and $2$ .

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Step 2.3.3.2.4.1

Factor $2$ out of $8$ .

$y + C_{1} = - 8 \int {u_{1}}^{3} e^{- {u_{1}}^{\frac{2 \cdot 4}{2}}} \frac{1}{2} d u_{1}$

Step 2.3.3.2.4.2

Cancel the common factors.

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Step 2.3.3.2.4.2.1

Factor $2$ out of $2$ .

$y + C_{1} = - 8 \int {u_{1}}^{3} e^{- {u_{1}}^{\frac{2 \cdot 4}{2 (1)}}} \frac{1}{2} d u_{1}$

Step 2.3.3.2.4.2.2

Cancel the common factor.

$y + C_{1} = - 8 \int {u_{1}}^{3} e^{- {u_{1}}^{\frac{2 \cdot 4}{2 \cdot 1}}} \frac{1}{2} d u_{1}$

Step 2.3.3.2.4.2.3

Rewrite the expression.

$y + C_{1} = - 8 \int {u_{1}}^{3} e^{- {u_{1}}^{\frac{4}{1}}} \frac{1}{2} d u_{1}$

Step 2.3.3.2.4.2.4

Divide $4$ by $1$ .

$y + C_{1} = - 8 \int {u_{1}}^{3} e^{- {u_{1}}^{4}} \frac{1}{2} d u_{1}$

Step 2.3.3.3

Combine $\frac{1}{2}$ and ${u_{1}}^{3}$ .

$y + C_{1} = - 8 \int \frac{{u_{1}}^{3}}{2} e^{- {u_{1}}^{4}} d u_{1}$

Step 2.3.3.4

Combine $\frac{{u_{1}}^{3}}{2}$ and $e^{- {u_{1}}^{4}}$ .

$y + C_{1} = - 8 \int \frac{{u_{1}}^{3} e^{- {u_{1}}^{4}}}{2} d u_{1}$

Step 2.3.4

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , move $\frac{1}{2}$ out of the integral.

$y + C_{1} = - 8 (\frac{1}{2} \int {u_{1}}^{3} e^{- {u_{1}}^{4}} d u_{1})$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Combine $\frac{1}{2}$ and $- 8$ .

$y + C_{1} = \frac{- 8}{2} \int {u_{1}}^{3} e^{- {u_{1}}^{4}} d u_{1}$

Step 2.3.5.2

Cancel the common factor of $- 8$ and $2$ .

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Step 2.3.5.2.1

Factor $2$ out of $- 8$ .

$y + C_{1} = \frac{2 \cdot - 4}{2} \int {u_{1}}^{3} e^{- {u_{1}}^{4}} d u_{1}$

Step 2.3.5.2.2

Cancel the common factors.

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Step 2.3.5.2.2.1

Factor $2$ out of $2$ .

$y + C_{1} = \frac{2 \cdot - 4}{2 (1)} \int {u_{1}}^{3} e^{- {u_{1}}^{4}} d u_{1}$

Step 2.3.5.2.2.2

Cancel the common factor.

$y + C_{1} = \frac{2 \cdot - 4}{2 \cdot 1} \int {u_{1}}^{3} e^{- {u_{1}}^{4}} d u_{1}$

Step 2.3.5.2.2.3

Rewrite the expression.

$y + C_{1} = \frac{- 4}{1} \int {u_{1}}^{3} e^{- {u_{1}}^{4}} d u_{1}$

Step 2.3.5.2.2.4

Divide $- 4$ by $1$ .

$y + C_{1} = - 4 \int {u_{1}}^{3} e^{- {u_{1}}^{4}} d u_{1}$

Step 2.3.6

Let $u_{2} = {u_{1}}^{2}$ . Then $d u_{2} = 2 u_{1} d u_{1}$ , so $\frac{1}{2} d u_{2} = u_{1} d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 2.3.6.1

Let $u_{2} = {u_{1}}^{2}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 2.3.6.1.1

Differentiate ${u_{1}}^{2}$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}]$

Step 2.3.6.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

$2 u_{1}$

Step 2.3.6.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$y + C_{1} = - 4 \int u_{2} e^{- {\sqrt{u_{2}}}^{4}} \frac{1}{2} d u_{2}$

Step 2.3.7

Simplify.

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Step 2.3.7.1

Rewrite ${\sqrt{u_{2}}}^{4}$ as ${u_{2}}^{2}$ .

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Step 2.3.7.1.1

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

$y + C_{1} = - 4 \int u_{2} e^{- {({u_{2}}^{\frac{1}{2}})}^{4}} \frac{1}{2} d u_{2}$

Step 2.3.7.1.2

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

$y + C_{1} = - 4 \int u_{2} e^{- {u_{2}}^{\frac{1}{2} \cdot 4}} \frac{1}{2} d u_{2}$

Step 2.3.7.1.3

Combine $\frac{1}{2}$ and $4$ .

$y + C_{1} = - 4 \int u_{2} e^{- {u_{2}}^{\frac{4}{2}}} \frac{1}{2} d u_{2}$

Step 2.3.7.1.4

Cancel the common factor of $4$ and $2$ .

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Step 2.3.7.1.4.1

Factor $2$ out of $4$ .

$y + C_{1} = - 4 \int u_{2} e^{- {u_{2}}^{\frac{2 \cdot 2}{2}}} \frac{1}{2} d u_{2}$

Step 2.3.7.1.4.2

Cancel the common factors.

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Step 2.3.7.1.4.2.1

Factor $2$ out of $2$ .

$y + C_{1} = - 4 \int u_{2} e^{- {u_{2}}^{\frac{2 \cdot 2}{2 (1)}}} \frac{1}{2} d u_{2}$

Step 2.3.7.1.4.2.2

Cancel the common factor.

$y + C_{1} = - 4 \int u_{2} e^{- {u_{2}}^{\frac{2 \cdot 2}{2 \cdot 1}}} \frac{1}{2} d u_{2}$

Step 2.3.7.1.4.2.3

Rewrite the expression.

$y + C_{1} = - 4 \int u_{2} e^{- {u_{2}}^{\frac{2}{1}}} \frac{1}{2} d u_{2}$

Step 2.3.7.1.4.2.4

Divide $2$ by $1$ .

$y + C_{1} = - 4 \int u_{2} e^{- {u_{2}}^{2}} \frac{1}{2} d u_{2}$

Step 2.3.7.2

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

$y + C_{1} = - 4 \int \frac{u_{2}}{2} e^{- {u_{2}}^{2}} d u_{2}$

Step 2.3.7.3

Combine $\frac{u_{2}}{2}$ and $e^{- {u_{2}}^{2}}$ .

$y + C_{1} = - 4 \int \frac{u_{2} e^{- {u_{2}}^{2}}}{2} d u_{2}$

Step 2.3.8

Since $\frac{1}{2}$ is constant with respect to $u_{2}$ , move $\frac{1}{2}$ out of the integral.

$y + C_{1} = - 4 (\frac{1}{2} \int u_{2} e^{- {u_{2}}^{2}} d u_{2})$

Step 2.3.9

Simplify.

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Step 2.3.9.1

Combine $\frac{1}{2}$ and $- 4$ .

$y + C_{1} = \frac{- 4}{2} \int u_{2} e^{- {u_{2}}^{2}} d u_{2}$

Step 2.3.9.2

Cancel the common factor of $- 4$ and $2$ .

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Step 2.3.9.2.1

Factor $2$ out of $- 4$ .

$y + C_{1} = \frac{2 \cdot - 2}{2} \int u_{2} e^{- {u_{2}}^{2}} d u_{2}$

Step 2.3.9.2.2

Cancel the common factors.

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Step 2.3.9.2.2.1

Factor $2$ out of $2$ .

$y + C_{1} = \frac{2 \cdot - 2}{2 (1)} \int u_{2} e^{- {u_{2}}^{2}} d u_{2}$

Step 2.3.9.2.2.2

Cancel the common factor.

$y + C_{1} = \frac{2 \cdot - 2}{2 \cdot 1} \int u_{2} e^{- {u_{2}}^{2}} d u_{2}$

Step 2.3.9.2.2.3

Rewrite the expression.

$y + C_{1} = \frac{- 2}{1} \int u_{2} e^{- {u_{2}}^{2}} d u_{2}$

Step 2.3.9.2.2.4

Divide $- 2$ by $1$ .

$y + C_{1} = - 2 \int u_{2} e^{- {u_{2}}^{2}} d u_{2}$

Step 2.3.10

Let $u_{3} = - {u_{2}}^{2}$ . Then $d u_{3} = - 2 u_{2} d u_{2}$ , so $- \frac{1}{2} d u_{3} = u_{2} d u_{2}$ . Rewrite using $u_{3}$ and $d$ $u_{3}$ .

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Step 2.3.10.1

Let $u_{3} = - {u_{2}}^{2}$ . Find $\frac{d u_{3}}{d u_{2}}$ .

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Step 2.3.10.1.1

Differentiate $- {u_{2}}^{2}$ .

$\frac{d}{d u_{2}} [- {u_{2}}^{2}]$

Step 2.3.10.1.2

Since $- 1$ is constant with respect to $u_{2}$ , the derivative of $- {u_{2}}^{2}$ with respect to $u_{2}$ is $- \frac{d}{d u_{2}} [{u_{2}}^{2}]$ .

$- \frac{d}{d u_{2}} [{u_{2}}^{2}]$

Step 2.3.10.1.3

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 2$ .

$- (2 u_{2})$

Step 2.3.10.1.4

Multiply $2$ by $- 1$ .

$- 2 u_{2}$

Step 2.3.10.2

Rewrite the problem using $u_{3}$ and $d u_{3}$ .

$y + C_{1} = - 2 \int e^{u_{3}} \frac{1}{- 2} d u_{3}$

Step 2.3.11

Simplify.

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Step 2.3.11.1

Move the negative in front of the fraction.

$y + C_{1} = - 2 \int e^{u_{3}} (- \frac{1}{2}) d u_{3}$

Step 2.3.11.2

Combine $e^{u_{3}}$ and $\frac{1}{2}$ .

$y + C_{1} = - 2 \int - \frac{e^{u_{3}}}{2} d u_{3}$

Step 2.3.12

Since $- 1$ is constant with respect to $u_{3}$ , move $- 1$ out of the integral.

$y + C_{1} = - 2 (- \int \frac{e^{u_{3}}}{2} d u_{3})$

Step 2.3.13

Multiply $- 1$ by $- 2$ .

$y + C_{1} = 2 \int \frac{e^{u_{3}}}{2} d u_{3}$

Step 2.3.14

Since $\frac{1}{2}$ is constant with respect to $u_{3}$ , move $\frac{1}{2}$ out of the integral.

$y + C_{1} = 2 (\frac{1}{2} \int e^{u_{3}} d u_{3})$

Step 2.3.15

Simplify.

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Step 2.3.15.1

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

$y + C_{1} = \frac{2}{2} \int e^{u_{3}} d u_{3}$

Step 2.3.15.2

Cancel the common factor of $2$ .

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Step 2.3.15.2.1

Cancel the common factor.

$y + C_{1} = \frac{2}{2} \int e^{u_{3}} d u_{3}$

Step 2.3.15.2.2

Rewrite the expression.

$y + C_{1} = 1 \int e^{u_{3}} d u_{3}$

Step 2.3.15.3

Multiply $\int e^{u_{3}} d u_{3}$ by $1$ .

$y + C_{1} = \int e^{u_{3}} d u_{3}$

Step 2.3.16

The integral of $e^{u_{3}}$ with respect to $u_{3}$ is $e^{u_{3}}$ .

$y + C_{1} = e^{u_{3}} + C_{2}$

Step 2.3.17

Substitute back in for each integration substitution variable.

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Step 2.3.17.1

Replace all occurrences of $u_{3}$ with $- {u_{2}}^{2}$ .

$y + C_{1} = e^{- {u_{2}}^{2}} + C_{2}$

Step 2.3.17.2

Replace all occurrences of $u_{2}$ with ${u_{1}}^{2}$ .

$y + C_{1} = e^{- {({u_{1}}^{2})}^{2}} + C_{2}$

Step 2.3.17.3

Replace all occurrences of $u_{1}$ with $x^{2}$ .

$y + C_{1} = e^{- {({(x^{2})}^{2})}^{2}} + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$y = e^{- {({(x^{2})}^{2})}^{2}} + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $8$ for $y$ in $y = e^{- {({(x^{2})}^{2})}^{2}} + K$ .

$8 = e^{- {({(0^{2})}^{2})}^{2}} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $e^{- {({(0^{2})}^{2})}^{2}} + K = 8$ .

$e^{- {({(0^{2})}^{2})}^{2}} + K = 8$

Step 4.2

Simplify each term.

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Step 4.2.1

Multiply the exponents in ${({(0^{2})}^{2})}^{2}$ .

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Step 4.2.1.1

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

$e^{- {(0^{2})}^{2 \cdot 2}} + K = 8$

Step 4.2.1.2

Multiply $2$ by $2$ .

$e^{- {(0^{2})}^{4}} + K = 8$

Step 4.2.2

Multiply the exponents in ${(0^{2})}^{4}$ .

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Step 4.2.2.1

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

$e^{- 0^{2 \cdot 4}} + K = 8$

Step 4.2.2.2

Multiply $2$ by $4$ .

$e^{- 0^{8}} + K = 8$

Step 4.2.3

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

$e^{- 0} + K = 8$

Step 4.2.4

Multiply $- 1$ by $0$ .

$e^{0} + K = 8$

Step 4.2.5

Anything raised to $0$ is $1$ .

$1 + K = 8$

Step 4.3

Move all terms not containing $K$ to the right side of the equation.

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Step 4.3.1

Subtract $1$ from both sides of the equation.

$K = 8 - 1$

Step 4.3.2

Subtract $1$ from $8$ .

$K = 7$

Step 5

Substitute $7$ for $K$ in $y = e^{- {({(x^{2})}^{2})}^{2}} + K$ and simplify.

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Step 5.1

Substitute $7$ for $K$ .

$y = e^{- {({(x^{2})}^{2})}^{2}} + 7$

Step 5.2

Simplify each term.

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Step 5.2.1

Multiply the exponents in ${({(x^{2})}^{2})}^{2}$ .

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Step 5.2.1.1

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

$y = e^{- {(x^{2})}^{2 \cdot 2}} + 7$

Step 5.2.1.2

Multiply $2$ by $2$ .

$y = e^{- {(x^{2})}^{4}} + 7$

Step 5.2.2

Multiply the exponents in ${(x^{2})}^{4}$ .

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Step 5.2.2.1

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

$y = e^{- x^{2 \cdot 4}} + 7$

Step 5.2.2.2

Multiply $2$ by $4$ .

$y = e^{- x^{8}} + 7$