Calculus Examples

$y = x^{\frac{3}{2}} - 3 x^{\frac{5}{2}}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

By the Sum Rule, the derivative of $x^{\frac{3}{2}} - 3 x^{\frac{5}{2}}$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{3}{2}}] + \frac{d}{d x} [- 3 x^{\frac{5}{2}}]$ .

$f' (x) = \frac{d}{d x} (x^{\frac{3}{2}}) + \frac{d}{d x} (- 3 x^{\frac{5}{2}})$

Step 1.1.2

Evaluate $\frac{d}{d x} [x^{\frac{3}{2}}]$ .

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Step 1.1.2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{2}$ .

$f' (x) = \frac{3}{2} x^{\frac{3}{2} - 1} + \frac{d}{d x} (- 3 x^{\frac{5}{2}})$

Step 1.1.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (x) = \frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d x} (- 3 x^{\frac{5}{2}})$

Step 1.1.2.3

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

$f' (x) = \frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d x} (- 3 x^{\frac{5}{2}})$

Step 1.1.2.4

Combine the numerators over the common denominator.

$f' (x) = \frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}} + \frac{d}{d x} (- 3 x^{\frac{5}{2}})$

Step 1.1.2.5

Simplify the numerator.

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Step 1.1.2.5.1

Multiply $- 1$ by $2$ .

$f' (x) = \frac{3}{2} x^{\frac{3 - 2}{2}} + \frac{d}{d x} (- 3 x^{\frac{5}{2}})$

Step 1.1.2.5.2

Subtract $2$ from $3$ .

$f' (x) = \frac{3}{2} x^{\frac{1}{2}} + \frac{d}{d x} (- 3 x^{\frac{5}{2}})$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 3 x^{\frac{5}{2}}]$ .

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Step 1.1.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{\frac{5}{2}}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{\frac{5}{2}}]$ .

$f' (x) = \frac{3}{2} x^{\frac{1}{2}} - 3 \frac{d}{d x} x^{\frac{5}{2}}$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{2}$ .

$f' (x) = \frac{3}{2} x^{\frac{1}{2}} - 3 (\frac{5}{2} x^{\frac{5}{2} - 1})$

Step 1.1.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (x) = \frac{3}{2} x^{\frac{1}{2}} - 3 (\frac{5}{2} x^{\frac{5}{2} - 1 \cdot \frac{2}{2}})$

Step 1.1.3.4

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

$f' (x) = \frac{3}{2} x^{\frac{1}{2}} - 3 (\frac{5}{2} x^{\frac{5}{2} + \frac{- 1 \cdot 2}{2}})$

Step 1.1.3.5

Combine the numerators over the common denominator.

$f' (x) = \frac{3}{2} x^{\frac{1}{2}} - 3 (\frac{5}{2} x^{\frac{5 - 1 \cdot 2}{2}})$

Step 1.1.3.6

Simplify the numerator.

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Step 1.1.3.6.1

Multiply $- 1$ by $2$ .

$f' (x) = \frac{3}{2} x^{\frac{1}{2}} - 3 (\frac{5}{2} x^{\frac{5 - 2}{2}})$

Step 1.1.3.6.2

Subtract $2$ from $5$ .

$f' (x) = \frac{3}{2} x^{\frac{1}{2}} - 3 (\frac{5}{2} x^{\frac{3}{2}})$

Step 1.1.3.7

Combine $\frac{5}{2}$ and $x^{\frac{3}{2}}$ .

$f' (x) = \frac{3}{2} x^{\frac{1}{2}} - 3 \frac{5 x^{\frac{3}{2}}}{2}$

Step 1.1.3.8

Combine $- 3$ and $\frac{5 x^{\frac{3}{2}}}{2}$ .

$f' (x) = \frac{3}{2} x^{\frac{1}{2}} + \frac{- 3 (5 x^{\frac{3}{2}})}{2}$

Step 1.1.3.9

Multiply $5$ by $- 3$ .

$f' (x) = \frac{3}{2} x^{\frac{1}{2}} + \frac{- 15 x^{\frac{3}{2}}}{2}$

Step 1.1.3.10

Move the negative in front of the fraction.

$f' (x) = \frac{3}{2} x^{\frac{1}{2}} - \frac{15 x^{\frac{3}{2}}}{2}$

Step 1.1.4

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

$f' (x) = \frac{3 x^{\frac{1}{2}}}{2} - \frac{15 x^{\frac{3}{2}}}{2}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{3 x^{\frac{1}{2}}}{2} - \frac{15 x^{\frac{3}{2}}}{2}$ .

$\frac{3 x^{\frac{1}{2}}}{2} - \frac{15 x^{\frac{3}{2}}}{2}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{3 x^{\frac{1}{2}}}{2} - \frac{15 x^{\frac{3}{2}}}{2} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{3 x^{\frac{1}{2}}}{2} - \frac{15 x^{\frac{3}{2}}}{2} = 0$

Step 2.2

Find a common factor $x^{\frac{1}{2}}$ that is present in each term.

$\frac{3 x^{\frac{1}{2}}}{2} - \frac{15}{2 {(x^{\frac{1}{2}})}^{3}}$

Step 2.3

Substitute $u$ for $x^{\frac{1}{2}}$ .

$\frac{3 (u)}{2} - \frac{15}{2 {(u)}^{3}} = 0$

Step 2.4

Solve for $u$ .

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Step 2.4.1

Factor each term.

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Step 2.4.1.1

Multiply $3$ by $u$ .

$\frac{3 u}{2} - \frac{15}{2 {(u)}^{3}} = 0$

Step 2.4.1.2

Remove parentheses.

$\frac{3 u}{2} - \frac{15}{2 u^{3}} = 0$

Step 2.4.2

Find the LCD of the terms in the equation.

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Step 2.4.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$2, 2 u^{3}, 1$

Step 2.4.2.2

Since $2, 2 u^{3}, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $2, 2, 1$ then find LCM for the variable part $u^{3}$ .

Step 2.4.2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.4.2.4

Since $2$ has no factors besides $1$ and $2$ .

$2$ is a prime number

Step 2.4.2.5

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.4.2.6

The LCM of $2, 2, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2$

Step 2.4.2.7

The factors for $u^{3}$ are $u \cdot u \cdot u$ , which is $u$ multiplied by each other $3$ times.

$u^{3} = u \cdot u \cdot u$

$u$ occurs $3$ times.

Step 2.4.2.8

The LCM of $u^{3}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$u \cdot u \cdot u$

Step 2.4.2.9

Simplify $u \cdot u \cdot u$ .

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Step 2.4.2.9.1

Multiply $u$ by $u$ .

$u^{2} \cdot u$

Step 2.4.2.9.2

Multiply $u^{2}$ by $u$ by adding the exponents.

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Step 2.4.2.9.2.1

Multiply $u^{2}$ by $u$ .

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Step 2.4.2.9.2.1.1

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

$u^{2} \cdot u^{1}$

Step 2.4.2.9.2.1.2

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

$u^{2 + 1}$

Step 2.4.2.9.2.2

Add $2$ and $1$ .

$u^{3}$

Step 2.4.2.10

The LCM for $2, 2 u^{3}, 1$ is the numeric part $2$ multiplied by the variable part.

$2 u^{3}$

Step 2.4.3

Multiply each term in $\frac{3 u}{2} - \frac{15}{2 u^{3}} = 0$ by $2 u^{3}$ to eliminate the fractions.

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Step 2.4.3.1

Multiply each term in $\frac{3 u}{2} - \frac{15}{2 u^{3}} = 0$ by $2 u^{3}$ .

$\frac{3 u}{2} (2 u^{3}) - \frac{15}{2 u^{3}} (2 u^{3}) = 0 (2 u^{3})$

Step 2.4.3.2

Simplify the left side.

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Step 2.4.3.2.1

Simplify each term.

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Step 2.4.3.2.1.1

Rewrite using the commutative property of multiplication.

$2 \frac{3 u}{2} u^{3} - \frac{15}{2 u^{3}} (2 u^{3}) = 0 (2 u^{3})$

Step 2.4.3.2.1.2

Cancel the common factor of $2$ .

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Step 2.4.3.2.1.2.1

Cancel the common factor.

$2 \frac{3 u}{2} u^{3} - \frac{15}{2 u^{3}} (2 u^{3}) = 0 (2 u^{3})$

Step 2.4.3.2.1.2.2

Rewrite the expression.

$3 u \cdot u^{3} - \frac{15}{2 u^{3}} (2 u^{3}) = 0 (2 u^{3})$

Step 2.4.3.2.1.3

Multiply $u$ by $u^{3}$ by adding the exponents.

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Step 2.4.3.2.1.3.1

Move $u^{3}$ .

$3 (u^{3} u) - \frac{15}{2 u^{3}} (2 u^{3}) = 0 (2 u^{3})$

Step 2.4.3.2.1.3.2

Multiply $u^{3}$ by $u$ .

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Step 2.4.3.2.1.3.2.1

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

$3 (u^{3} u^{1}) - \frac{15}{2 u^{3}} (2 u^{3}) = 0 (2 u^{3})$

Step 2.4.3.2.1.3.2.2

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

$3 u^{3 + 1} - \frac{15}{2 u^{3}} (2 u^{3}) = 0 (2 u^{3})$

Step 2.4.3.2.1.3.3

Add $3$ and $1$ .

$3 u^{4} - \frac{15}{2 u^{3}} (2 u^{3}) = 0 (2 u^{3})$

Step 2.4.3.2.1.4

Cancel the common factor of $2 u^{3}$ .

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Step 2.4.3.2.1.4.1

Move the leading negative in $- \frac{15}{2 u^{3}}$ into the numerator.

$3 u^{4} + \frac{- 15}{2 u^{3}} (2 u^{3}) = 0 (2 u^{3})$

Step 2.4.3.2.1.4.2

Cancel the common factor.

$3 u^{4} + \frac{- 15}{2 u^{3}} (2 u^{3}) = 0 (2 u^{3})$

Step 2.4.3.2.1.4.3

Rewrite the expression.

$3 u^{4} - 15 = 0 (2 u^{3})$

Step 2.4.3.3

Simplify the right side.

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Step 2.4.3.3.1

Multiply $0 (2 u^{3})$ .

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Step 2.4.3.3.1.1

Multiply $2$ by $0$ .

$3 u^{4} - 15 = 0 u^{3}$

Step 2.4.3.3.1.2

Multiply $0$ by $u^{3}$ .

$3 u^{4} - 15 = 0$

Step 2.4.4

Solve the equation.

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Step 2.4.4.1

Add $15$ to both sides of the equation.

$3 u^{4} = 15$

Step 2.4.4.2

Divide each term in $3 u^{4} = 15$ by $3$ and simplify.

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Step 2.4.4.2.1

Divide each term in $3 u^{4} = 15$ by $3$ .

$\frac{3 u^{4}}{3} = \frac{15}{3}$

Step 2.4.4.2.2

Simplify the left side.

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Step 2.4.4.2.2.1

Cancel the common factor of $3$ .

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Step 2.4.4.2.2.1.1

Cancel the common factor.

$\frac{3 u^{4}}{3} = \frac{15}{3}$

Step 2.4.4.2.2.1.2

Divide $u^{4}$ by $1$ .

$u^{4} = \frac{15}{3}$

Step 2.4.4.2.3

Simplify the right side.

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Step 2.4.4.2.3.1

Divide $15$ by $3$ .

$u^{4} = 5$

Step 2.4.4.3

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

$u = \pm \sqrt[4]{5}$

Step 2.4.4.4

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

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Step 2.4.4.4.1

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

$u = \sqrt[4]{5}$

Step 2.4.4.4.2

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

$u = - \sqrt[4]{5}$

Step 2.4.4.4.3

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

$u = \sqrt[4]{5}, - \sqrt[4]{5}$

Step 2.5

Substitute $x$ for $u$ .

$x^{\frac{1}{2}} = \sqrt[4]{5}, - \sqrt[4]{5}$

Step 2.6

Solve for $x^{\frac{1}{2}} = \sqrt[4]{5}$ for $x$ .

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Step 2.6.1

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{2}})}^{2} = {\sqrt[4]{5}}^{2}$

Step 2.6.2

Simplify the exponent.

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Step 2.6.2.1

Simplify the left side.

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Step 2.6.2.1.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

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Step 2.6.2.1.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 2.6.2.1.1.1.1

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

$x^{\frac{1}{2} \cdot 2} = {\sqrt[4]{5}}^{2}$

Step 2.6.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 2.6.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {\sqrt[4]{5}}^{2}$

Step 2.6.2.1.1.1.2.2

Rewrite the expression.

$x^{1} = {\sqrt[4]{5}}^{2}$

Step 2.6.2.1.1.2

Simplify.

$x = {\sqrt[4]{5}}^{2}$

Step 2.6.2.2

Simplify the right side.

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Step 2.6.2.2.1

Rewrite ${\sqrt[4]{5}}^{2}$ as $\sqrt{5}$ .

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Step 2.6.2.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{5}$ as $5^{\frac{1}{4}}$ .

$x = {(5^{\frac{1}{4}})}^{2}$

Step 2.6.2.2.1.2

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

$x = 5^{\frac{1}{4} \cdot 2}$

Step 2.6.2.2.1.3

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

$x = 5^{\frac{2}{4}}$

Step 2.6.2.2.1.4

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

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Step 2.6.2.2.1.4.1

Factor $2$ out of $2$ .

$x = 5^{\frac{2 (1)}{4}}$

Step 2.6.2.2.1.4.2

Cancel the common factors.

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Step 2.6.2.2.1.4.2.1

Factor $2$ out of $4$ .

$x = 5^{\frac{2 \cdot 1}{2 \cdot 2}}$

Step 2.6.2.2.1.4.2.2

Cancel the common factor.

$x = 5^{\frac{2 \cdot 1}{2 \cdot 2}}$

Step 2.6.2.2.1.4.2.3

Rewrite the expression.

$x = 5^{\frac{1}{2}}$

Step 2.6.2.2.1.5

Rewrite $5^{\frac{1}{2}}$ as $\sqrt{5}$ .

$x = \sqrt{5}$

Step 2.7

List all of the solutions.

$x = \sqrt{5}, \sqrt{5}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Convert expressions with fractional exponents to radicals.

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Step 3.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{3 \sqrt{x^{1}}}{2} - \frac{15 x^{\frac{3}{2}}}{2}$

Step 3.1.2

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{3 \sqrt{x^{1}}}{2} - \frac{15 \sqrt{x^{3}}}{2}$

Step 3.1.3

Anything raised to $1$ is the base itself.

$\frac{3 \sqrt{x}}{2} - \frac{15 \sqrt{x^{3}}}{2}$

Step 3.2

Set the radicand in $\sqrt{x}$ less than $0$ to find where the expression is undefined.

$x < 0$

Step 3.3

Set the radicand in $\sqrt{x^{3}}$ less than $0$ to find where the expression is undefined.

$x^{3} < 0$

Step 3.4

Solve for $x$ .

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Step 3.4.1

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

$\sqrt[3]{x^{3}} < \sqrt[3]{0}$

Step 3.4.2

Simplify the equation.

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Step 3.4.2.1

Simplify the left side.

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Step 3.4.2.1.1

Pull terms out from under the radical.

$x < \sqrt[3]{0}$

Step 3.4.2.2

Simplify the right side.

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Step 3.4.2.2.1

Simplify $\sqrt[3]{0}$ .

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Step 3.4.2.2.1.1

Rewrite $0$ as $0^{3}$ .

$x < \sqrt[3]{0^{3}}$

Step 3.4.2.2.1.2

Pull terms out from under the radical.

$x < 0$

Step 3.5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x < 0$

$(- \infty, 0)$

$x < 0$

$(- \infty, 0)$

Step 4

Evaluate $x^{\frac{3}{2}} - 3 x^{\frac{5}{2}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \sqrt{5}$ .

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Step 4.1.1

Substitute $\sqrt{5}$ for $x$ .

${(\sqrt{5})}^{\frac{3}{2}} - 3 {(\sqrt{5})}^{\frac{5}{2}}$

Step 4.1.2

Remove parentheses.

${\sqrt{5}}^{\frac{3}{2}} - 3 {\sqrt{5}}^{\frac{5}{2}}$

Step 4.2

List all of the points.

$(\sqrt{5}, {\sqrt{5}}^{\frac{3}{2}} - 3 {\sqrt{5}}^{\frac{5}{2}})$

Step 5