Calculus Examples

$y = x^{\frac{9}{8}}$ , $y = 7 x^{\frac{1}{8}}$

Step 1

Solve by substitution to find the intersection between the curves.

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Step 1.1

Eliminate the equal sides of each equation and combine.

$x^{\frac{9}{8}} = 7 x^{\frac{1}{8}}$

Step 1.2

Solve $x^{\frac{9}{8}} = 7 x^{\frac{1}{8}}$ for $x$ .

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Step 1.2.1

Eliminate the fractional exponents by multiplying both exponents by the LCD.

${(x^{\frac{9}{8}})}^{8} = {(7 x^{\frac{1}{8}})}^{8}$

Step 1.2.2

Multiply the exponents in ${(x^{\frac{9}{8}})}^{8}$ .

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Step 1.2.2.1

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

$x^{\frac{9}{8} \cdot 8} = {(7 x^{\frac{1}{8}})}^{8}$

Step 1.2.2.2

Cancel the common factor of $8$ .

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Step 1.2.2.2.1

Cancel the common factor.

$x^{\frac{9}{8} \cdot 8} = {(7 x^{\frac{1}{8}})}^{8}$

Step 1.2.2.2.2

Rewrite the expression.

$x^{9} = {(7 x^{\frac{1}{8}})}^{8}$

Step 1.2.3

Simplify ${(7 x^{\frac{1}{8}})}^{8}$ .

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Step 1.2.3.1

Apply the product rule to $7 x^{\frac{1}{8}}$ .

$x^{9} = 7^{8} {(x^{\frac{1}{8}})}^{8}$

Step 1.2.3.2

Raise $7$ to the power of $8$ .

$x^{9} = 5764801 {(x^{\frac{1}{8}})}^{8}$

Step 1.2.3.3

Multiply the exponents in ${(x^{\frac{1}{8}})}^{8}$ .

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Step 1.2.3.3.1

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

$x^{9} = 5764801 x^{\frac{1}{8} \cdot 8}$

Step 1.2.3.3.2

Cancel the common factor of $8$ .

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Step 1.2.3.3.2.1

Cancel the common factor.

$x^{9} = 5764801 x^{\frac{1}{8} \cdot 8}$

Step 1.2.3.3.2.2

Rewrite the expression.

$x^{9} = 5764801 x^{1}$

Step 1.2.3.4

Simplify.

$x^{9} = 5764801 x$

Step 1.2.4

Subtract $5764801 x$ from both sides of the equation.

$x^{9} - 5764801 x = 0$

Step 1.2.5

Factor the left side of the equation.

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Step 1.2.5.1

Factor $x$ out of $x^{9} - 5764801 x$ .

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Step 1.2.5.1.1

Factor $x$ out of $x^{9}$ .

$x \cdot x^{8} - 5764801 x = 0$

Step 1.2.5.1.2

Factor $x$ out of $- 5764801 x$ .

$x \cdot x^{8} + x \cdot - 5764801 = 0$

Step 1.2.5.1.3

Factor $x$ out of $x \cdot x^{8} + x \cdot - 5764801$ .

$x (x^{8} - 5764801) = 0$

Step 1.2.5.2

Rewrite $x^{8}$ as ${(x^{4})}^{2}$ .

$x ({(x^{4})}^{2} - 5764801) = 0$

Step 1.2.5.3

Rewrite $5764801$ as $2401^{2}$ .

$x ({(x^{4})}^{2} - 2401^{2}) = 0$

Step 1.2.5.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{4}$ and $b = 2401$ .

$x ((x^{4} + 2401) (x^{4} - 2401)) = 0$

Step 1.2.5.5

Factor.

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Step 1.2.5.5.1

Simplify.

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Step 1.2.5.5.1.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$x ((x^{4} + 2401) ({(x^{2})}^{2} - 2401)) = 0$

Step 1.2.5.5.1.2

Rewrite $2401$ as $49^{2}$ .

$x ((x^{4} + 2401) ({(x^{2})}^{2} - 49^{2})) = 0$

Step 1.2.5.5.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 49$ .

$x ((x^{4} + 2401) ((x^{2} + 49) (x^{2} - 49))) = 0$

Step 1.2.5.5.1.4

Factor.

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Step 1.2.5.5.1.4.1

Simplify.

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Step 1.2.5.5.1.4.1.1

Rewrite $49$ as $7^{2}$ .

$x ((x^{4} + 2401) ((x^{2} + 49) (x^{2} - 7^{2}))) = 0$

Step 1.2.5.5.1.4.1.2

Factor.

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Step 1.2.5.5.1.4.1.2.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 7$ .

$x ((x^{4} + 2401) ((x^{2} + 49) ((x + 7) (x - 7)))) = 0$

Step 1.2.5.5.1.4.1.2.2

Remove unnecessary parentheses.

$x ((x^{4} + 2401) ((x^{2} + 49) (x + 7) (x - 7))) = 0$

Step 1.2.5.5.1.4.2

Remove unnecessary parentheses.

$x ((x^{4} + 2401) (x^{2} + 49) (x + 7) (x - 7)) = 0$

Step 1.2.5.5.2

Remove unnecessary parentheses.

$x (x^{4} + 2401) (x^{2} + 49) (x + 7) (x - 7) = 0$

Step 1.2.6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x^{4} + 2401 = 0$

$x^{2} + 49 = 0$

$x + 7 = 0$

$x - 7 = 0 + y = 7 x^{\frac{1}{8}}$

Step 1.2.7

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.8

Set $x^{4} + 2401$ equal to $0$ and solve for $x$ .

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Step 1.2.8.1

Set $x^{4} + 2401$ equal to $0$ .

$x^{4} + 2401 = 0$

Step 1.2.8.2

Solve $x^{4} + 2401 = 0$ for $x$ .

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Step 1.2.8.2.1

Subtract $2401$ from both sides of the equation.

$x^{4} = - 2401$

Step 1.2.8.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[4]{- 2401}$

Step 1.2.8.2.3

Simplify $\pm \sqrt[4]{- 2401}$ .

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Step 1.2.8.2.3.1

Rewrite $- 2401$ as $7^{4} \cdot - 1$ .

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Step 1.2.8.2.3.1.1

Factor $2401$ out of $- 2401$ .

$x = \pm \sqrt[4]{2401 (- 1)}$

Step 1.2.8.2.3.1.2

Rewrite $2401$ as $7^{4}$ .

$x = \pm \sqrt[4]{7^{4} \cdot - 1}$

Step 1.2.8.2.3.2

Pull terms out from under the radical.

$x = \pm 7 \sqrt[4]{- 1}$

Step 1.2.8.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.8.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 7 \sqrt[4]{- 1}$

Step 1.2.8.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 7 \sqrt[4]{- 1}$

Step 1.2.8.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 7 \sqrt[4]{- 1}, - 7 \sqrt[4]{- 1}$

Step 1.2.9

Set $x^{2} + 49$ equal to $0$ and solve for $x$ .

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Step 1.2.9.1

Set $x^{2} + 49$ equal to $0$ .

$x^{2} + 49 = 0$

Step 1.2.9.2

Solve $x^{2} + 49 = 0$ for $x$ .

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Step 1.2.9.2.1

Subtract $49$ from both sides of the equation.

$x^{2} = - 49$

Step 1.2.9.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 49}$

Step 1.2.9.2.3

Simplify $\pm \sqrt{- 49}$ .

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Step 1.2.9.2.3.1

Rewrite $- 49$ as $- 1 (49)$ .

$x = \pm \sqrt{- 1 (49)}$

Step 1.2.9.2.3.2

Rewrite $\sqrt{- 1 (49)}$ as $\sqrt{- 1} \cdot \sqrt{49}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{49}$

Step 1.2.9.2.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \cdot \sqrt{49}$

Step 1.2.9.2.3.4

Rewrite $49$ as $7^{2}$ .

$x = \pm i \cdot \sqrt{7^{2}}$

Step 1.2.9.2.3.5

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

$x = \pm i \cdot 7$

Step 1.2.9.2.3.6

Move $7$ to the left of $i$ .

$x = \pm 7 i$

Step 1.2.9.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.9.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 7 i$

Step 1.2.9.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 7 i$

Step 1.2.9.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 7 i, - 7 i$

Step 1.2.10

Set $x + 7$ equal to $0$ and solve for $x$ .

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Step 1.2.10.1

Set $x + 7$ equal to $0$ .

$x + 7 = 0$

Step 1.2.10.2

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 1.2.11

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 1.2.11.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 1.2.11.2

Add $7$ to both sides of the equation.

$x = 7$

Step 1.2.12

The final solution is all the values that make $x (x^{4} + 2401) (x^{2} + 49) (x + 7) (x - 7) = 0$ true.

$x = 0, 7 \sqrt[4]{- 1}, - 7 \sqrt[4]{- 1}, 7 i, - 7 i, - 7, 7$

Step 1.3

Evaluate $y$ when $x = 0$ .

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Step 1.3.1

Substitute $0$ for $x$ .

$y = 7 {(0)}^{\frac{1}{8}}$

Step 1.3.2

Substitute $0$ for $x$ in $y = 7 {(0)}^{\frac{1}{8}}$ and solve for $y$ .

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Step 1.3.2.1

Remove parentheses.

$y = 7 \cdot 0^{\frac{1}{8}}$

Step 1.3.2.2

Simplify $7 \cdot 0^{\frac{1}{8}}$ .

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Step 1.3.2.2.1

Simplify the expression.

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Step 1.3.2.2.1.1

Rewrite $0$ as $0^{8}$ .

$y = 7 \cdot {(0^{8})}^{\frac{1}{8}}$

Step 1.3.2.2.1.2

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

$y = 7 \cdot 0^{8 (\frac{1}{8})}$

Step 1.3.2.2.2

Cancel the common factor of $8$ .

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Step 1.3.2.2.2.1

Cancel the common factor.

$y = 7 \cdot 0^{8 (\frac{1}{8})}$

Step 1.3.2.2.2.2

Rewrite the expression.

$y = 7 \cdot 0^{1}$

Step 1.3.2.2.3

Evaluate the exponent.

$y = 7 \cdot 0$

Step 1.3.2.2.4

Multiply $7$ by $0$ .

$y = 0$

Step 1.4

Evaluate $y$ when $x = 7 \sqrt[4]{- 1}$ .

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Step 1.4.1

Substitute $7 \sqrt[4]{- 1}$ for $x$ .

$y = 7 {(7 \sqrt[4]{- 1})}^{\frac{1}{8}}$

Step 1.4.2

Simplify $7 {(7 \sqrt[4]{- 1})}^{\frac{1}{8}}$ .

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Step 1.4.2.1

Apply the product rule to $7 \sqrt[4]{- 1}$ .

$y = 7 (7^{\frac{1}{8}} {\sqrt[4]{- 1}}^{\frac{1}{8}})$

Step 1.4.2.2

Multiply $7$ by $7^{\frac{1}{8}}$ by adding the exponents.

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Step 1.4.2.2.1

Move $7^{\frac{1}{8}}$ .

$y = 7^{\frac{1}{8}} \cdot 7 {\sqrt[4]{- 1}}^{\frac{1}{8}}$

Step 1.4.2.2.2

Multiply $7^{\frac{1}{8}}$ by $7$ .

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Step 1.4.2.2.2.1

Raise $7$ to the power of $1$ .

$y = 7^{\frac{1}{8}} \cdot 7^{1} {\sqrt[4]{- 1}}^{\frac{1}{8}}$

Step 1.4.2.2.2.2

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

$y = 7^{\frac{1}{8} + 1} {\sqrt[4]{- 1}}^{\frac{1}{8}}$

Step 1.4.2.2.3

Write $1$ as a fraction with a common denominator.

$y = 7^{\frac{1}{8} + \frac{8}{8}} {\sqrt[4]{- 1}}^{\frac{1}{8}}$

Step 1.4.2.2.4

Combine the numerators over the common denominator.

$y = 7^{\frac{1 + 8}{8}} {\sqrt[4]{- 1}}^{\frac{1}{8}}$

Step 1.4.2.2.5

Add $1$ and $8$ .

$y = 7^{\frac{9}{8}} {\sqrt[4]{- 1}}^{\frac{1}{8}}$

Step 1.5

Evaluate $y$ when $x = - 7 \sqrt[4]{- 1}$ .

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Step 1.5.1

Substitute $- 7 \sqrt[4]{- 1}$ for $x$ .

$y = 7 {(- 7 \sqrt[4]{- 1})}^{\frac{1}{8}}$

Step 1.5.2

Apply the product rule to $- 7 \sqrt[4]{- 1}$ .

$y = 7 {(- 7)}^{\frac{1}{8}} {\sqrt[4]{- 1}}^{\frac{1}{8}}$

Step 1.6

Evaluate $y$ when $x = 7 i$ .

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Step 1.6.1

Substitute $7 i$ for $x$ .

$y = 7 {(7 i)}^{\frac{1}{8}}$

Step 1.6.2

Simplify $7 {(7 i)}^{\frac{1}{8}}$ .

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Step 1.6.2.1

Apply the product rule to $7 i$ .

$y = 7 (7^{\frac{1}{8}} i^{\frac{1}{8}})$

Step 1.6.2.2

Multiply $7$ by $7^{\frac{1}{8}}$ by adding the exponents.

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Step 1.6.2.2.1

Move $7^{\frac{1}{8}}$ .

$y = 7^{\frac{1}{8}} \cdot 7 i^{\frac{1}{8}}$

Step 1.6.2.2.2

Multiply $7^{\frac{1}{8}}$ by $7$ .

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Step 1.6.2.2.2.1

Raise $7$ to the power of $1$ .

$y = 7^{\frac{1}{8}} \cdot 7^{1} i^{\frac{1}{8}}$

Step 1.6.2.2.2.2

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

$y = 7^{\frac{1}{8} + 1} i^{\frac{1}{8}}$

Step 1.6.2.2.3

Write $1$ as a fraction with a common denominator.

$y = 7^{\frac{1}{8} + \frac{8}{8}} i^{\frac{1}{8}}$

Step 1.6.2.2.4

Combine the numerators over the common denominator.

$y = 7^{\frac{1 + 8}{8}} i^{\frac{1}{8}}$

Step 1.6.2.2.5

Add $1$ and $8$ .

$y = 7^{\frac{9}{8}} i^{\frac{1}{8}}$

Step 1.7

Evaluate $y$ when $x = - 7 i$ .

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Step 1.7.1

Substitute $- 7 i$ for $x$ .

$y = 7 {(- 7 i)}^{\frac{1}{8}}$

Step 1.7.2

Apply the product rule to $- 7 i$ .

$y = 7 {(- 7)}^{\frac{1}{8}} i^{\frac{1}{8}}$

Step 1.8

Substitute $- 7$ for $x$ .

$y = 7 {(- 7)}^{\frac{1}{8}}$

Step 1.9

Evaluate $y$ when $x = 7$ .

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Step 1.9.1

Substitute $7$ for $x$ .

$y = 7 {(7)}^{\frac{1}{8}}$

Step 1.9.2

Substitute $7$ for $x$ in $y = 7 {(7)}^{\frac{1}{8}}$ and solve for $y$ .

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Step 1.9.2.1

Remove parentheses.

$y = 7 \cdot 7^{\frac{1}{8}}$

Step 1.9.2.2

Multiply $7$ by $7^{\frac{1}{8}}$ by adding the exponents.

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Step 1.9.2.2.1

Multiply $7$ by $7^{\frac{1}{8}}$ .

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Step 1.9.2.2.1.1

Raise $7$ to the power of $1$ .

$y = 7^{1} \cdot 7^{\frac{1}{8}}$

Step 1.9.2.2.1.2

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

$y = 7^{1 + \frac{1}{8}}$

Step 1.9.2.2.2

Write $1$ as a fraction with a common denominator.

$y = 7^{\frac{8}{8} + \frac{1}{8}}$

Step 1.9.2.2.3

Combine the numerators over the common denominator.

$y = 7^{\frac{8 + 1}{8}}$

Step 1.9.2.2.4

Add $8$ and $1$ .

$y = 7^{\frac{9}{8}}$

Step 1.10

List all of the solutions.

$y = 0, x = 0$

$y = 7^{\frac{9}{8}} {\sqrt[4]{- 1}}^{\frac{1}{8}}, x = 7 \sqrt[4]{- 1}$

$y = 7 {(- 7)}^{\frac{1}{8}} {\sqrt[4]{- 1}}^{\frac{1}{8}}, x = - 7 \sqrt[4]{- 1}$

$y = 7^{\frac{9}{8}} i^{\frac{1}{8}}, x = 7 i$

$y = 7 {(- 7)}^{\frac{1}{8}} i^{\frac{1}{8}}, x = - 7 i$

$y = 7 {(- 7)}^{\frac{1}{8}}, x = - 7$

$y = 7^{\frac{9}{8}}, x = 7$

$y = 0, x = 0$

$y = 7^{\frac{9}{8}} {\sqrt[4]{- 1}}^{\frac{1}{8}}, x = 7 \sqrt[4]{- 1}$

$y = 7 {(- 7)}^{\frac{1}{8}} {\sqrt[4]{- 1}}^{\frac{1}{8}}, x = - 7 \sqrt[4]{- 1}$

$y = 7^{\frac{9}{8}} i^{\frac{1}{8}}, x = 7 i$

$y = 7 {(- 7)}^{\frac{1}{8}} i^{\frac{1}{8}}, x = - 7 i$

$y = 7 {(- 7)}^{\frac{1}{8}}, x = - 7$

$y = 7^{\frac{9}{8}}, x = 7$

Step 2

The area between the given curves is unbounded.

Unbounded area

Step 3