Calculus Examples

$y = x^{\frac{5}{4}}$ , $y = 3 x^{\frac{1}{4}}$

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.

$x^{\frac{5}{4}} = 3 x^{\frac{1}{4}}$

Step 1.2

Solve $x^{\frac{5}{4}} = 3 x^{\frac{1}{4}}$ for $x$ .

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Step 1.2.1

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

${(x^{\frac{5}{4}})}^{4} = {(3 x^{\frac{1}{4}})}^{4}$

Step 1.2.2

Multiply the exponents in ${(x^{\frac{5}{4}})}^{4}$ .

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Step 1.2.2.1

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

$x^{\frac{5}{4} \cdot 4} = {(3 x^{\frac{1}{4}})}^{4}$

Step 1.2.2.2

Cancel the common factor of $4$ .

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Step 1.2.2.2.1

Cancel the common factor.

$x^{\frac{5}{4} \cdot 4} = {(3 x^{\frac{1}{4}})}^{4}$

Step 1.2.2.2.2

Rewrite the expression.

$x^{5} = {(3 x^{\frac{1}{4}})}^{4}$

Step 1.2.3

Simplify ${(3 x^{\frac{1}{4}})}^{4}$ .

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Step 1.2.3.1

Apply the product rule to $3 x^{\frac{1}{4}}$ .

$x^{5} = 3^{4} {(x^{\frac{1}{4}})}^{4}$

Step 1.2.3.2

Raise $3$ to the power of $4$ .

$x^{5} = 81 {(x^{\frac{1}{4}})}^{4}$

Step 1.2.3.3

Multiply the exponents in ${(x^{\frac{1}{4}})}^{4}$ .

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Step 1.2.3.3.1

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

$x^{5} = 81 x^{\frac{1}{4} \cdot 4}$

Step 1.2.3.3.2

Cancel the common factor of $4$ .

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Step 1.2.3.3.2.1

Cancel the common factor.

$x^{5} = 81 x^{\frac{1}{4} \cdot 4}$

Step 1.2.3.3.2.2

Rewrite the expression.

$x^{5} = 81 x^{1}$

Step 1.2.3.4

Simplify.

$x^{5} = 81 x$

Step 1.2.4

Subtract $81 x$ from both sides of the equation.

$x^{5} - 81 x = 0$

Step 1.2.5

Factor the left side of the equation.

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Step 1.2.5.1

Factor $x$ out of $x^{5} - 81 x$ .

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Step 1.2.5.1.1

Factor $x$ out of $x^{5}$ .

$x \cdot x^{4} - 81 x = 0$

Step 1.2.5.1.2

Factor $x$ out of $- 81 x$ .

$x \cdot x^{4} + x \cdot - 81 = 0$

Step 1.2.5.1.3

Factor $x$ out of $x \cdot x^{4} + x \cdot - 81$ .

$x (x^{4} - 81) = 0$

Step 1.2.5.2

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

$x ({(x^{2})}^{2} - 81) = 0$

Step 1.2.5.3

Rewrite $81$ as $9^{2}$ .

$x ({(x^{2})}^{2} - 9^{2}) = 0$

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 $a = x^{2}$ and $b = 9$ .

$x ((x^{2} + 9) (x^{2} - 9)) = 0$

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

$x ((x^{2} + 9) (x^{2} - 3^{2})) = 0$

Step 1.2.5.5.1.2

Factor.

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Step 1.2.5.5.1.2.1

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 $b = 3$ .

$x ((x^{2} + 9) ((x + 3) (x - 3))) = 0$

Step 1.2.5.5.1.2.2

Remove unnecessary parentheses.

$x ((x^{2} + 9) (x + 3) (x - 3)) = 0$

Step 1.2.5.5.2

Remove unnecessary parentheses.

$x (x^{2} + 9) (x + 3) (x - 3) = 0$

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 $0$ .

$x = 0$

$x^{2} + 9 = 0$

$x + 3 = 0$

$x - 3 = 0 + y = 3 x^{\frac{1}{4}}$

Step 1.2.7

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.8

Set $x^{2} + 9$ equal to $0$ and solve for $x$ .

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Step 1.2.8.1

Set $x^{2} + 9$ equal to $0$ .

$x^{2} + 9 = 0$

Step 1.2.8.2

Solve $x^{2} + 9 = 0$ for $x$ .

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Step 1.2.8.2.1

Subtract $9$ from both sides of the equation.

$x^{2} = - 9$

Step 1.2.8.2.2

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

$x = \pm \sqrt{- 9}$

Step 1.2.8.2.3

Simplify $\pm \sqrt{- 9}$ .

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Step 1.2.8.2.3.1

Rewrite $- 9$ as $- 1 (9)$ .

$x = \pm \sqrt{- 1 (9)}$

Step 1.2.8.2.3.2

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{9}$

Step 1.2.8.2.3.3

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

$x = \pm i \cdot \sqrt{9}$

Step 1.2.8.2.3.4

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

$x = \pm i \cdot \sqrt{3^{2}}$

Step 1.2.8.2.3.5

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

$x = \pm i \cdot 3$

Step 1.2.8.2.3.6

Move $3$ to the left of $i$ .

$x = \pm 3 i$

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.

$x = 3 i$

Step 1.2.8.2.4.2

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

$x = - 3 i$

Step 1.2.8.2.4.3

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

$x = 3 i, - 3 i$

Step 1.2.9

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 1.2.9.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 1.2.9.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 1.2.10

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 1.2.10.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 1.2.10.2

Add $3$ to both sides of the equation.

$x = 3$

Step 1.2.11

The final solution is all the values that make $x (x^{2} + 9) (x + 3) (x - 3) = 0$ true.

$x = 0, 3 i, - 3 i, - 3, 3$

Step 1.3

Evaluate $y$ when $x = 0$ .

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Step 1.3.1

Substitute $0$ for $x$ .

$y = 3 {(0)}^{\frac{1}{4}}$

Step 1.3.2

Substitute $0$ for $x$ in $y = 3 {(0)}^{\frac{1}{4}}$ and solve for $y$ .

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Step 1.3.2.1

Remove parentheses.

$y = 3 \cdot 0^{\frac{1}{4}}$

Step 1.3.2.2

Simplify $3 \cdot 0^{\frac{1}{4}}$ .

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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 $0^{4}$ .

$y = 3 \cdot {(0^{4})}^{\frac{1}{4}}$

Step 1.3.2.2.1.2

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

$y = 3 \cdot 0^{4 (\frac{1}{4})}$

Step 1.3.2.2.2

Cancel the common factor of $4$ .

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Step 1.3.2.2.2.1

Cancel the common factor.

$y = 3 \cdot 0^{4 (\frac{1}{4})}$

Step 1.3.2.2.2.2

Rewrite the expression.

$y = 3 \cdot 0^{1}$

Step 1.3.2.2.3

Evaluate the exponent.

$y = 3 \cdot 0$

Step 1.3.2.2.4

Multiply $3$ by $0$ .

$y = 0$

Step 1.4

Evaluate $y$ when $x = 3 i$ .

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Step 1.4.1

Substitute $3 i$ for $x$ .

$y = 3 {(3 i)}^{\frac{1}{4}}$

Step 1.4.2

Simplify $3 {(3 i)}^{\frac{1}{4}}$ .

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Step 1.4.2.1

Apply the product rule to $3 i$ .

$y = 3 (3^{\frac{1}{4}} i^{\frac{1}{4}})$

Step 1.4.2.2

Multiply $3$ by $3^{\frac{1}{4}}$ by adding the exponents.

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Step 1.4.2.2.1

Move $3^{\frac{1}{4}}$ .

$y = 3^{\frac{1}{4}} \cdot 3 i^{\frac{1}{4}}$

Step 1.4.2.2.2

Multiply $3^{\frac{1}{4}}$ by $3$ .

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Step 1.4.2.2.2.1

Raise $3$ to the power of $1$ .

$y = 3^{\frac{1}{4}} \cdot 3^{1} i^{\frac{1}{4}}$

Step 1.4.2.2.2.2

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

$y = 3^{\frac{1}{4} + 1} i^{\frac{1}{4}}$

Step 1.4.2.2.3

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

$y = 3^{\frac{1}{4} + \frac{4}{4}} i^{\frac{1}{4}}$

Step 1.4.2.2.4

Combine the numerators over the common denominator.

$y = 3^{\frac{1 + 4}{4}} i^{\frac{1}{4}}$

Step 1.4.2.2.5

Add $1$ and $4$ .

$y = 3^{\frac{5}{4}} i^{\frac{1}{4}}$

Step 1.5

Evaluate $y$ when $x = - 3 i$ .

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Step 1.5.1

Substitute $- 3 i$ for $x$ .

$y = 3 {(- 3 i)}^{\frac{1}{4}}$

Step 1.5.2

Apply the product rule to $- 3 i$ .

$y = 3 {(- 3)}^{\frac{1}{4}} i^{\frac{1}{4}}$

Step 1.6

Substitute $- 3$ for $x$ .

$y = 3 {(- 3)}^{\frac{1}{4}}$

Step 1.7

Evaluate $y$ when $x = 3$ .

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Step 1.7.1

Substitute $3$ for $x$ .

$y = 3 {(3)}^{\frac{1}{4}}$

Step 1.7.2

Substitute $3$ for $x$ in $y = 3 {(3)}^{\frac{1}{4}}$ and solve for $y$ .

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Step 1.7.2.1

Remove parentheses.

$y = 3 \cdot 3^{\frac{1}{4}}$

Step 1.7.2.2

Multiply $3$ by $3^{\frac{1}{4}}$ by adding the exponents.

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Step 1.7.2.2.1

Multiply $3$ by $3^{\frac{1}{4}}$ .

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Step 1.7.2.2.1.1

Raise $3$ to the power of $1$ .

$y = 3^{1} \cdot 3^{\frac{1}{4}}$

Step 1.7.2.2.1.2

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

$y = 3^{1 + \frac{1}{4}}$

Step 1.7.2.2.2

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

$y = 3^{\frac{4}{4} + \frac{1}{4}}$

Step 1.7.2.2.3

Combine the numerators over the common denominator.

$y = 3^{\frac{4 + 1}{4}}$

Step 1.7.2.2.4

Add $4$ and $1$ .

$y = 3^{\frac{5}{4}}$

Step 1.8

List all of the solutions.

$y = 0, x = 0$

$y = 3^{\frac{5}{4}} i^{\frac{1}{4}}, x = 3 i$

$y = 3 {(- 3)}^{\frac{1}{4}} i^{\frac{1}{4}}, x = - 3 i$

$y = 3 {(- 3)}^{\frac{1}{4}}, x = - 3$

$y = 3^{\frac{5}{4}}, x = 3$

$y = 0, x = 0$

$y = 3^{\frac{5}{4}} i^{\frac{1}{4}}, x = 3 i$

$y = 3 {(- 3)}^{\frac{1}{4}} i^{\frac{1}{4}}, x = - 3 i$

$y = 3 {(- 3)}^{\frac{1}{4}}, x = - 3$

$y = 3^{\frac{5}{4}}, x = 3$

Step 2

The area between the given curves is unbounded.

Unbounded area

Step 3