Calculus Examples

$f (4) = 4 x^{\frac{5}{4}} + 4 x^{\frac{3}{2}} + 2 x$

Step 1

Replace the variable $x$ with $4$ in the expression.

$f (4) = 4 {(4)}^{\frac{5}{4}} + 4 {(4)}^{\frac{3}{2}} + 2 (4)$

Step 2

Simplify the result.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $4$ by ${(4)}^{\frac{5}{4}}$ by adding the exponents.

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Step 2.1.1.1

Multiply $4$ by ${(4)}^{\frac{5}{4}}$ .

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Step 2.1.1.1.1

Raise $4$ to the power of $1$ .

$f (4) = 4 {(4)}^{\frac{5}{4}} + 4 {(4)}^{\frac{3}{2}} + 2 (4)$

Step 2.1.1.1.2

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

$f (4) = 4^{1 + \frac{5}{4}} + 4 {(4)}^{\frac{3}{2}} + 2 (4)$

Step 2.1.1.2

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

$f (4) = 4^{\frac{4}{4} + \frac{5}{4}} + 4 {(4)}^{\frac{3}{2}} + 2 (4)$

Step 2.1.1.3

Combine the numerators over the common denominator.

$f (4) = 4^{\frac{4 + 5}{4}} + 4 {(4)}^{\frac{3}{2}} + 2 (4)$

Step 2.1.1.4

Add $4$ and $5$ .

$f (4) = 4^{\frac{9}{4}} + 4 {(4)}^{\frac{3}{2}} + 2 (4)$

Step 2.1.2

Multiply $4$ by ${(4)}^{\frac{3}{2}}$ by adding the exponents.

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Step 2.1.2.1

Multiply $4$ by ${(4)}^{\frac{3}{2}}$ .

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Step 2.1.2.1.1

Raise $4$ to the power of $1$ .

$f (4) = 4^{\frac{9}{4}} + 4 {(4)}^{\frac{3}{2}} + 2 (4)$

Step 2.1.2.1.2

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

$f (4) = 4^{\frac{9}{4}} + 4^{1 + \frac{3}{2}} + 2 (4)$

Step 2.1.2.2

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

$f (4) = 4^{\frac{9}{4}} + 4^{\frac{2}{2} + \frac{3}{2}} + 2 (4)$

Step 2.1.2.3

Combine the numerators over the common denominator.

$f (4) = 4^{\frac{9}{4}} + 4^{\frac{2 + 3}{2}} + 2 (4)$

Step 2.1.2.4

Add $2$ and $3$ .

$f (4) = 4^{\frac{9}{4}} + 4^{\frac{5}{2}} + 2 (4)$

Step 2.1.3

Rewrite $4$ as $2^{2}$ .

$f (4) = 4^{\frac{9}{4}} + {(2^{2})}^{\frac{5}{2}} + 2 (4)$

Step 2.1.4

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

$f (4) = 4^{\frac{9}{4}} + 2^{2 (\frac{5}{2})} + 2 (4)$

Step 2.1.5

Cancel the common factor of $2$ .

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Step 2.1.5.1

Cancel the common factor.

$f (4) = 4^{\frac{9}{4}} + 2^{2 (\frac{5}{2})} + 2 (4)$

Step 2.1.5.2

Rewrite the expression.

$f (4) = 4^{\frac{9}{4}} + 2^{5} + 2 (4)$

Step 2.1.6

Raise $2$ to the power of $5$ .

$f (4) = 4^{\frac{9}{4}} + 32 + 2 (4)$

Step 2.1.7

Multiply $2$ by $4$ .

$f (4) = 4^{\frac{9}{4}} + 32 + 8$

Step 2.2

Add $32$ and $8$ .

$f (4) = 4^{\frac{9}{4}} + 40$

Step 2.3

The final answer is $4^{\frac{9}{4}} + 40$ .

$4^{\frac{9}{4}} + 40$

Step 3

The result can be shown in multiple forms.

Exact Form:

$4^{\frac{9}{4}} + 40$

Decimal Form:

$62.62741699 \dots$

Step 4