Calculus Examples

$g (t) = (2 t + 3 \sqrt{t} + 5) (\sqrt{t} + 4)$

Step 1

Apply basic rules of exponents.

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Step 1.1

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

$\frac{d}{d t} [(2 t + 3 t^{\frac{1}{2}} + 5) (\sqrt{t} + 4)]$

Step 1.2

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

$\frac{d}{d t} [(2 t + 3 t^{\frac{1}{2}} + 5) (t^{\frac{1}{2}} + 4)]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = 2 t + 3 t^{\frac{1}{2}} + 5$ and $g (t) = t^{\frac{1}{2}} + 4$ .

$(2 t + 3 t^{\frac{1}{2}} + 5) \frac{d}{d t} [t^{\frac{1}{2}} + 4] + (t^{\frac{1}{2}} + 4) \frac{d}{d t} [2 t + 3 t^{\frac{1}{2}} + 5]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $t^{\frac{1}{2}} + 4$ with respect to $t$ is $\frac{d}{d t} [t^{\frac{1}{2}}] + \frac{d}{d t} [4]$ .

$(2 t + 3 t^{\frac{1}{2}} + 5) (\frac{d}{d t} [t^{\frac{1}{2}}] + \frac{d}{d t} [4]) + (t^{\frac{1}{2}} + 4) \frac{d}{d t} [2 t + 3 t^{\frac{1}{2}} + 5]$

Step 3.2

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

$(2 t + 3 t^{\frac{1}{2}} + 5) (\frac{1}{2} t^{\frac{1}{2} - 1} + \frac{d}{d t} [4]) + (t^{\frac{1}{2}} + 4) \frac{d}{d t} [2 t + 3 t^{\frac{1}{2}} + 5]$

Step 4

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

$(2 t + 3 t^{\frac{1}{2}} + 5) (\frac{1}{2} t^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d t} [4]) + (t^{\frac{1}{2}} + 4) \frac{d}{d t} [2 t + 3 t^{\frac{1}{2}} + 5]$

Step 5

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

$(2 t + 3 t^{\frac{1}{2}} + 5) (\frac{1}{2} t^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d t} [4]) + (t^{\frac{1}{2}} + 4) \frac{d}{d t} [2 t + 3 t^{\frac{1}{2}} + 5]$

Step 6

Combine the numerators over the common denominator.

$(2 t + 3 t^{\frac{1}{2}} + 5) (\frac{1}{2} t^{\frac{1 - 1 \cdot 2}{2}} + \frac{d}{d t} [4]) + (t^{\frac{1}{2}} + 4) \frac{d}{d t} [2 t + 3 t^{\frac{1}{2}} + 5]$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

$(2 t + 3 t^{\frac{1}{2}} + 5) (\frac{1}{2} t^{\frac{1 - 2}{2}} + \frac{d}{d t} [4]) + (t^{\frac{1}{2}} + 4) \frac{d}{d t} [2 t + 3 t^{\frac{1}{2}} + 5]$

Step 7.2

Subtract $2$ from $1$ .

$(2 t + 3 t^{\frac{1}{2}} + 5) (\frac{1}{2} t^{\frac{- 1}{2}} + \frac{d}{d t} [4]) + (t^{\frac{1}{2}} + 4) \frac{d}{d t} [2 t + 3 t^{\frac{1}{2}} + 5]$

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

$(2 t + 3 t^{\frac{1}{2}} + 5) (\frac{1}{2} t^{- \frac{1}{2}} + \frac{d}{d t} [4]) + (t^{\frac{1}{2}} + 4) \frac{d}{d t} [2 t + 3 t^{\frac{1}{2}} + 5]$

Step 8.2

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

$(2 t + 3 t^{\frac{1}{2}} + 5) (\frac{t^{- \frac{1}{2}}}{2} + \frac{d}{d t} [4]) + (t^{\frac{1}{2}} + 4) \frac{d}{d t} [2 t + 3 t^{\frac{1}{2}} + 5]$

Step 8.3

Move $t^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$(2 t + 3 t^{\frac{1}{2}} + 5) (\frac{1}{2 t^{\frac{1}{2}}} + \frac{d}{d t} [4]) + (t^{\frac{1}{2}} + 4) \frac{d}{d t} [2 t + 3 t^{\frac{1}{2}} + 5]$

Step 9

Since $4$ is constant with respect to $t$ , the derivative of $4$ with respect to $t$ is $0$ .

$(2 t + 3 t^{\frac{1}{2}} + 5) (\frac{1}{2 t^{\frac{1}{2}}} + 0) + (t^{\frac{1}{2}} + 4) \frac{d}{d t} [2 t + 3 t^{\frac{1}{2}} + 5]$

Step 10

Add $\frac{1}{2 t^{\frac{1}{2}}}$ and $0$ .

$(2 t + 3 t^{\frac{1}{2}} + 5) \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) \frac{d}{d t} [2 t + 3 t^{\frac{1}{2}} + 5]$

Step 11

By the Sum Rule, the derivative of $2 t + 3 t^{\frac{1}{2}} + 5$ with respect to $t$ is $\frac{d}{d t} [2 t] + \frac{d}{d t} [3 t^{\frac{1}{2}}] + \frac{d}{d t} [5]$ .

$(2 t + 3 t^{\frac{1}{2}} + 5) \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) (\frac{d}{d t} [2 t] + \frac{d}{d t} [3 t^{\frac{1}{2}}] + \frac{d}{d t} [5])$

Step 12

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

$(2 t + 3 t^{\frac{1}{2}} + 5) \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) (2 \frac{d}{d t} [t] + \frac{d}{d t} [3 t^{\frac{1}{2}}] + \frac{d}{d t} [5])$

Step 13

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

$(2 t + 3 t^{\frac{1}{2}} + 5) \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) (2 \cdot 1 + \frac{d}{d t} [3 t^{\frac{1}{2}}] + \frac{d}{d t} [5])$

Step 14

Multiply $2$ by $1$ .

$(2 t + 3 t^{\frac{1}{2}} + 5) \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) (2 + \frac{d}{d t} [3 t^{\frac{1}{2}}] + \frac{d}{d t} [5])$

Step 15

Since $3$ is constant with respect to $t$ , the derivative of $3 t^{\frac{1}{2}}$ with respect to $t$ is $3 \frac{d}{d t} [t^{\frac{1}{2}}]$ .

$(2 t + 3 t^{\frac{1}{2}} + 5) \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) (2 + 3 \frac{d}{d t} [t^{\frac{1}{2}}] + \frac{d}{d t} [5])$

Step 16

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

$(2 t + 3 t^{\frac{1}{2}} + 5) \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) (2 + 3 (\frac{1}{2} t^{\frac{1}{2} - 1}) + \frac{d}{d t} [5])$

Step 17

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

$(2 t + 3 t^{\frac{1}{2}} + 5) \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) (2 + 3 (\frac{1}{2} t^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d t} [5])$

Step 18

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

$(2 t + 3 t^{\frac{1}{2}} + 5) \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) (2 + 3 (\frac{1}{2} t^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d t} [5])$

Step 19

Combine the numerators over the common denominator.

$(2 t + 3 t^{\frac{1}{2}} + 5) \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) (2 + 3 (\frac{1}{2} t^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d t} [5])$

Step 20

Simplify the numerator.

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Step 20.1

Multiply $- 1$ by $2$ .

$(2 t + 3 t^{\frac{1}{2}} + 5) \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) (2 + 3 (\frac{1}{2} t^{\frac{1 - 2}{2}}) + \frac{d}{d t} [5])$

Step 20.2

Subtract $2$ from $1$ .

$(2 t + 3 t^{\frac{1}{2}} + 5) \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) (2 + 3 (\frac{1}{2} t^{\frac{- 1}{2}}) + \frac{d}{d t} [5])$

Step 21

Combine fractions.

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Step 21.1

Move the negative in front of the fraction.

$(2 t + 3 t^{\frac{1}{2}} + 5) \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) (2 + 3 (\frac{1}{2} t^{- \frac{1}{2}}) + \frac{d}{d t} [5])$

Step 21.2

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

$(2 t + 3 t^{\frac{1}{2}} + 5) \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) (2 + 3 \frac{t^{- \frac{1}{2}}}{2} + \frac{d}{d t} [5])$

Step 21.3

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

$(2 t + 3 t^{\frac{1}{2}} + 5) \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) (2 + \frac{3 t^{- \frac{1}{2}}}{2} + \frac{d}{d t} [5])$

Step 21.4

Move $t^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$(2 t + 3 t^{\frac{1}{2}} + 5) \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) (2 + \frac{3}{2 t^{\frac{1}{2}}} + \frac{d}{d t} [5])$

Step 22

Since $5$ is constant with respect to $t$ , the derivative of $5$ with respect to $t$ is $0$ .

$(2 t + 3 t^{\frac{1}{2}} + 5) \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) (2 + \frac{3}{2 t^{\frac{1}{2}}} + 0)$

Step 23

Add $2 + \frac{3}{2 t^{\frac{1}{2}}}$ and $0$ .

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

Step 24

Simplify.

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Step 24.1

Apply the distributive property.

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

Step 24.2

Combine terms.

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Step 24.2.1

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

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

Step 24.2.2

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

$\frac{t \cdot 2}{2 t^{\frac{1}{2}}} + 3 t^{\frac{1}{2}} \frac{1}{2 t^{\frac{1}{2}}} + 5 \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) (2 + \frac{3}{2 t^{\frac{1}{2}}})$

Step 24.2.3

Move $2$ to the left of $t$ .

$\frac{2 \cdot t}{2 t^{\frac{1}{2}}} + 3 t^{\frac{1}{2}} \frac{1}{2 t^{\frac{1}{2}}} + 5 \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) (2 + \frac{3}{2 t^{\frac{1}{2}}})$

Step 24.2.4

Move $t^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{2 t \cdot t^{- \frac{1}{2}}}{2} + 3 t^{\frac{1}{2}} \frac{1}{2 t^{\frac{1}{2}}} + 5 \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) (2 + \frac{3}{2 t^{\frac{1}{2}}})$

Step 24.2.5

Multiply $t$ by $t^{- \frac{1}{2}}$ by adding the exponents.

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Step 24.2.5.1

Move $t^{- \frac{1}{2}}$ .

$\frac{2 (t^{- \frac{1}{2}} t)}{2} + 3 t^{\frac{1}{2}} \frac{1}{2 t^{\frac{1}{2}}} + 5 \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) (2 + \frac{3}{2 t^{\frac{1}{2}}})$

Step 24.2.5.2

Multiply $t^{- \frac{1}{2}}$ by $t$ .

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Step 24.2.5.2.1

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

$\frac{2 (t^{- \frac{1}{2}} t^{1})}{2} + 3 t^{\frac{1}{2}} \frac{1}{2 t^{\frac{1}{2}}} + 5 \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) (2 + \frac{3}{2 t^{\frac{1}{2}}})$

Step 24.2.5.2.2

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

$\frac{2 t^{- \frac{1}{2} + 1}}{2} + 3 t^{\frac{1}{2}} \frac{1}{2 t^{\frac{1}{2}}} + 5 \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) (2 + \frac{3}{2 t^{\frac{1}{2}}})$

Step 24.2.5.3

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

$\frac{2 t^{- \frac{1}{2} + \frac{2}{2}}}{2} + 3 t^{\frac{1}{2}} \frac{1}{2 t^{\frac{1}{2}}} + 5 \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) (2 + \frac{3}{2 t^{\frac{1}{2}}})$

Step 24.2.5.4

Combine the numerators over the common denominator.

$\frac{2 t^{\frac{- 1 + 2}{2}}}{2} + 3 t^{\frac{1}{2}} \frac{1}{2 t^{\frac{1}{2}}} + 5 \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) (2 + \frac{3}{2 t^{\frac{1}{2}}})$

Step 24.2.5.5

Add $- 1$ and $2$ .

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

Step 24.2.6

Cancel the common factor.

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

Step 24.2.7

Divide $t^{\frac{1}{2}}$ by $1$ .

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

Step 24.2.8

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

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

Step 24.2.9

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

$t^{\frac{1}{2}} + \frac{t^{\frac{1}{2}} \cdot 3}{2 t^{\frac{1}{2}}} + 5 \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) (2 + \frac{3}{2 t^{\frac{1}{2}}})$

Step 24.2.10

Move $3$ to the left of $t^{\frac{1}{2}}$ .

$t^{\frac{1}{2}} + \frac{3 \cdot t^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} + 5 \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} + 4) (2 + \frac{3}{2 t^{\frac{1}{2}}})$

Step 24.2.11

Cancel the common factor.

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

Step 24.2.12

Rewrite the expression.

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

Step 24.2.13

Combine $5$ and $\frac{1}{2 t^{\frac{1}{2}}}$ .

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

Step 24.2.14

To write $(t^{\frac{1}{2}} + 4) (2 + \frac{3}{2 t^{\frac{1}{2}}})$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$t^{\frac{1}{2}} + \frac{3}{2} + (t^{\frac{1}{2}} + 4) (2 + \frac{3}{2 t^{\frac{1}{2}}}) \cdot \frac{2}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.2.15

Combine $(t^{\frac{1}{2}} + 4) (2 + \frac{3}{2 t^{\frac{1}{2}}})$ and $\frac{2}{2}$ .

$t^{\frac{1}{2}} + \frac{3}{2} + \frac{(t^{\frac{1}{2}} + 4) (2 + \frac{3}{2 t^{\frac{1}{2}}}) \cdot 2}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.2.16

Combine the numerators over the common denominator.

$t^{\frac{1}{2}} + \frac{3 + (t^{\frac{1}{2}} + 4) (2 + \frac{3}{2 t^{\frac{1}{2}}}) \cdot 2}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.2.17

Move $2$ to the left of $(t^{\frac{1}{2}} + 4) (2 + \frac{3}{2 t^{\frac{1}{2}}})$ .

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

Step 24.3

Simplify each term.

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Step 24.3.1

Simplify the numerator.

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Step 24.3.1.1

Apply the distributive property.

$t^{\frac{1}{2}} + \frac{3 + (2 t^{\frac{1}{2}} + 2 \cdot 4) (2 + \frac{3}{2 t^{\frac{1}{2}}})}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.2

Multiply $2$ by $4$ .

$t^{\frac{1}{2}} + \frac{3 + (2 t^{\frac{1}{2}} + 8) (2 + \frac{3}{2 t^{\frac{1}{2}}})}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.3

Expand $(2 t^{\frac{1}{2}} + 8) (2 + \frac{3}{2 t^{\frac{1}{2}}})$ using the FOIL Method.

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Step 24.3.1.3.1

Apply the distributive property.

$t^{\frac{1}{2}} + \frac{3 + 2 t^{\frac{1}{2}} (2 + \frac{3}{2 t^{\frac{1}{2}}}) + 8 (2 + \frac{3}{2 t^{\frac{1}{2}}})}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.3.2

Apply the distributive property.

$t^{\frac{1}{2}} + \frac{3 + 2 t^{\frac{1}{2}} \cdot 2 + 2 t^{\frac{1}{2}} \frac{3}{2 t^{\frac{1}{2}}} + 8 (2 + \frac{3}{2 t^{\frac{1}{2}}})}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.3.3

Apply the distributive property.

$t^{\frac{1}{2}} + \frac{3 + 2 t^{\frac{1}{2}} \cdot 2 + 2 t^{\frac{1}{2}} \frac{3}{2 t^{\frac{1}{2}}} + 8 \cdot 2 + 8 \frac{3}{2 t^{\frac{1}{2}}}}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.4

Simplify and combine like terms.

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Step 24.3.1.4.1

Simplify each term.

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Step 24.3.1.4.1.1

Multiply $2$ by $2$ .

$t^{\frac{1}{2}} + \frac{3 + 4 t^{\frac{1}{2}} + 2 t^{\frac{1}{2}} \frac{3}{2 t^{\frac{1}{2}}} + 8 \cdot 2 + 8 \frac{3}{2 t^{\frac{1}{2}}}}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.4.1.2

Cancel the common factor of $2 t^{\frac{1}{2}}$ .

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Step 24.3.1.4.1.2.1

Cancel the common factor.

$t^{\frac{1}{2}} + \frac{3 + 4 t^{\frac{1}{2}} + 2 t^{\frac{1}{2}} \frac{3}{2 t^{\frac{1}{2}}} + 8 \cdot 2 + 8 \frac{3}{2 t^{\frac{1}{2}}}}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.4.1.2.2

Rewrite the expression.

$t^{\frac{1}{2}} + \frac{3 + 4 t^{\frac{1}{2}} + 3 + 8 \cdot 2 + 8 \frac{3}{2 t^{\frac{1}{2}}}}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.4.1.3

Multiply $8$ by $2$ .

$t^{\frac{1}{2}} + \frac{3 + 4 t^{\frac{1}{2}} + 3 + 16 + 8 \frac{3}{2 t^{\frac{1}{2}}}}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.4.1.4

Cancel the common factor of $2$ .

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Step 24.3.1.4.1.4.1

Factor $2$ out of $8$ .

$t^{\frac{1}{2}} + \frac{3 + 4 t^{\frac{1}{2}} + 3 + 16 + 2 \cdot 4 \frac{3}{2 t^{\frac{1}{2}}}}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.4.1.4.2

Cancel the common factor.

$t^{\frac{1}{2}} + \frac{3 + 4 t^{\frac{1}{2}} + 3 + 16 + 2 \cdot 4 \frac{3}{2 t^{\frac{1}{2}}}}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.4.1.4.3

Rewrite the expression.

$t^{\frac{1}{2}} + \frac{3 + 4 t^{\frac{1}{2}} + 3 + 16 + 4 \frac{3}{t^{\frac{1}{2}}}}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.4.1.5

Combine $4$ and $\frac{3}{t^{\frac{1}{2}}}$ .

$t^{\frac{1}{2}} + \frac{3 + 4 t^{\frac{1}{2}} + 3 + 16 + \frac{4 \cdot 3}{t^{\frac{1}{2}}}}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.4.1.6

Multiply $4$ by $3$ .

$t^{\frac{1}{2}} + \frac{3 + 4 t^{\frac{1}{2}} + 3 + 16 + \frac{12}{t^{\frac{1}{2}}}}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.4.2

Add $3$ and $16$ .

$t^{\frac{1}{2}} + \frac{3 + 4 t^{\frac{1}{2}} + 19 + \frac{12}{t^{\frac{1}{2}}}}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.5

Add $3$ and $19$ .

$t^{\frac{1}{2}} + \frac{4 t^{\frac{1}{2}} + 22 + \frac{12}{t^{\frac{1}{2}}}}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.6

Factor $2$ out of $4 t^{\frac{1}{2}} + 22 + \frac{12}{t^{\frac{1}{2}}}$ .

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Step 24.3.1.6.1

Factor $2$ out of $4 t^{\frac{1}{2}}$ .

$t^{\frac{1}{2}} + \frac{2 (2 t^{\frac{1}{2}}) + 22 + \frac{12}{t^{\frac{1}{2}}}}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.6.2

Factor $2$ out of $22$ .

$t^{\frac{1}{2}} + \frac{2 (2 t^{\frac{1}{2}}) + 2 (11) + \frac{12}{t^{\frac{1}{2}}}}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.6.3

Factor $2$ out of $\frac{12}{t^{\frac{1}{2}}}$ .

$t^{\frac{1}{2}} + \frac{2 (2 t^{\frac{1}{2}}) + 2 \cdot 11 + 2 \frac{6}{t^{\frac{1}{2}}}}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.6.4

Factor $2$ out of $2 (2 t^{\frac{1}{2}}) + 2 \cdot 11$ .

$t^{\frac{1}{2}} + \frac{2 (2 t^{\frac{1}{2}} + 11) + 2 \frac{6}{t^{\frac{1}{2}}}}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.6.5

Factor $2$ out of $2 (2 t^{\frac{1}{2}} + 11) + 2 \frac{6}{t^{\frac{1}{2}}}$ .

$t^{\frac{1}{2}} + \frac{2 (2 t^{\frac{1}{2}} + 11 + \frac{6}{t^{\frac{1}{2}}})}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.7

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

$t^{\frac{1}{2}} + \frac{2 (\frac{2 t^{\frac{1}{2}} t^{\frac{1}{2}}}{t^{\frac{1}{2}}} + \frac{6}{t^{\frac{1}{2}}} + 11)}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.8

Combine the numerators over the common denominator.

$t^{\frac{1}{2}} + \frac{2 (\frac{2 t^{\frac{1}{2}} t^{\frac{1}{2}} + 6}{t^{\frac{1}{2}}} + 11)}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.9

Simplify the numerator.

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Step 24.3.1.9.1

Factor $2$ out of $2 t^{\frac{1}{2}} t^{\frac{1}{2}} + 6$ .

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Step 24.3.1.9.1.1

Factor $2$ out of $2 t^{\frac{1}{2}} t^{\frac{1}{2}}$ .

$t^{\frac{1}{2}} + \frac{2 (\frac{2 (t^{\frac{1}{2}} t^{\frac{1}{2}}) + 6}{t^{\frac{1}{2}}} + 11)}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.9.1.2

Factor $2$ out of $6$ .

$t^{\frac{1}{2}} + \frac{2 (\frac{2 (t^{\frac{1}{2}} t^{\frac{1}{2}}) + 2 \cdot 3}{t^{\frac{1}{2}}} + 11)}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.9.1.3

Factor $2$ out of $2 (t^{\frac{1}{2}} t^{\frac{1}{2}}) + 2 \cdot 3$ .

$t^{\frac{1}{2}} + \frac{2 (\frac{2 (t^{\frac{1}{2}} t^{\frac{1}{2}} + 3)}{t^{\frac{1}{2}}} + 11)}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.9.2

Multiply $t^{\frac{1}{2}}$ by $t^{\frac{1}{2}}$ by adding the exponents.

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Step 24.3.1.9.2.1

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

$t^{\frac{1}{2}} + \frac{2 (\frac{2 (t^{\frac{1}{2} + \frac{1}{2}} + 3)}{t^{\frac{1}{2}}} + 11)}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.9.2.2

Combine the numerators over the common denominator.

$t^{\frac{1}{2}} + \frac{2 (\frac{2 (t^{\frac{1 + 1}{2}} + 3)}{t^{\frac{1}{2}}} + 11)}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.9.2.3

Add $1$ and $1$ .

$t^{\frac{1}{2}} + \frac{2 (\frac{2 (t^{\frac{2}{2}} + 3)}{t^{\frac{1}{2}}} + 11)}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.9.2.4

Divide $2$ by $2$ .

$t^{\frac{1}{2}} + \frac{2 (\frac{2 (t^{1} + 3)}{t^{\frac{1}{2}}} + 11)}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.9.3

Simplify $t^{1}$ .

$t^{\frac{1}{2}} + \frac{2 (\frac{2 (t + 3)}{t^{\frac{1}{2}}} + 11)}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.10

To write $11$ as a fraction with a common denominator, multiply by $\frac{t^{\frac{1}{2}}}{t^{\frac{1}{2}}}$ .

$t^{\frac{1}{2}} + \frac{2 (\frac{2 (t + 3)}{t^{\frac{1}{2}}} + \frac{11 t^{\frac{1}{2}}}{t^{\frac{1}{2}}})}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.11

Combine the numerators over the common denominator.

$t^{\frac{1}{2}} + \frac{2 \frac{2 (t + 3) + 11 t^{\frac{1}{2}}}{t^{\frac{1}{2}}}}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.12

Simplify the numerator.

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Step 24.3.1.12.1

Apply the distributive property.

$t^{\frac{1}{2}} + \frac{2 \frac{2 t + 2 \cdot 3 + 11 t^{\frac{1}{2}}}{t^{\frac{1}{2}}}}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.12.2

Multiply $2$ by $3$ .

$t^{\frac{1}{2}} + \frac{2 \frac{2 t + 6 + 11 t^{\frac{1}{2}}}{t^{\frac{1}{2}}}}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.1.12.3

Reorder terms.

$t^{\frac{1}{2}} + \frac{2 \frac{2 t + 11 t^{\frac{1}{2}} + 6}{t^{\frac{1}{2}}}}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.2

Combine $2$ and $\frac{2 t + 11 t^{\frac{1}{2}} + 6}{t^{\frac{1}{2}}}$ .

$t^{\frac{1}{2}} + \frac{\frac{2 (2 t + 11 t^{\frac{1}{2}} + 6)}{t^{\frac{1}{2}}}}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.3

Multiply the numerator by the reciprocal of the denominator.

$t^{\frac{1}{2}} + \frac{2 (2 t + 11 t^{\frac{1}{2}} + 6)}{t^{\frac{1}{2}}} \cdot \frac{1}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.4

Combine.

$t^{\frac{1}{2}} + \frac{2 (2 t + 11 t^{\frac{1}{2}} + 6) \cdot 1}{t^{\frac{1}{2}} \cdot 2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.5

Cancel the common factor.

$t^{\frac{1}{2}} + \frac{2 (2 t + 11 t^{\frac{1}{2}} + 6) \cdot 1}{t^{\frac{1}{2}} \cdot 2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.6

Rewrite the expression.

$t^{\frac{1}{2}} + \frac{(2 t + 11 t^{\frac{1}{2}} + 6) \cdot 1}{t^{\frac{1}{2}}} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.3.7

Multiply $2 t + 11 t^{\frac{1}{2}} + 6$ by $1$ .

$t^{\frac{1}{2}} + \frac{2 t + 11 t^{\frac{1}{2}} + 6}{t^{\frac{1}{2}}} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.4

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

$\frac{t^{\frac{1}{2}} t^{\frac{1}{2}}}{t^{\frac{1}{2}}} + \frac{2 t + 11 t^{\frac{1}{2}} + 6}{t^{\frac{1}{2}}} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.5

Combine the numerators over the common denominator.

$\frac{t^{\frac{1}{2}} t^{\frac{1}{2}} + 2 t + 11 t^{\frac{1}{2}} + 6}{t^{\frac{1}{2}}} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.6

Simplify the numerator.

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Step 24.6.1

Multiply $t^{\frac{1}{2}}$ by $t^{\frac{1}{2}}$ by adding the exponents.

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Step 24.6.1.1

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

$\frac{t^{\frac{1}{2} + \frac{1}{2}} + 2 t + 11 t^{\frac{1}{2}} + 6}{t^{\frac{1}{2}}} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.6.1.2

Combine the numerators over the common denominator.

$\frac{t^{\frac{1 + 1}{2}} + 2 t + 11 t^{\frac{1}{2}} + 6}{t^{\frac{1}{2}}} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.6.1.3

Add $1$ and $1$ .

$\frac{t^{\frac{2}{2}} + 2 t + 11 t^{\frac{1}{2}} + 6}{t^{\frac{1}{2}}} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.6.1.4

Divide $2$ by $2$ .

$\frac{t^{1} + 2 t + 11 t^{\frac{1}{2}} + 6}{t^{\frac{1}{2}}} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.6.2

Simplify $t^{1}$ .

$\frac{t + 2 t + 11 t^{\frac{1}{2}} + 6}{t^{\frac{1}{2}}} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.6.3

Add $t$ and $2 t$ .

$\frac{3 t + 11 t^{\frac{1}{2}} + 6}{t^{\frac{1}{2}}} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.6.4

Rewrite $3 t + 11 t^{\frac{1}{2}} + 6$ in a factored form.

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Step 24.6.4.1

Rewrite $t$ as ${(t^{\frac{1}{2}})}^{2}$ .

$\frac{3 {(t^{\frac{1}{2}})}^{2} + 11 t^{\frac{1}{2}} + 6}{t^{\frac{1}{2}}} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.6.4.2

Let $u = t^{\frac{1}{2}}$ . Substitute $u$ for all occurrences of $t^{\frac{1}{2}}$ .

$\frac{3 u^{2} + 11 u + 6}{t^{\frac{1}{2}}} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.6.4.3

Factor by grouping.

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Step 24.6.4.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 6 = 18$ and whose sum is $b = 11$ .

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Step 24.6.4.3.1.1

Factor $11$ out of $11 u$ .

$\frac{3 u^{2} + 11 (u) + 6}{t^{\frac{1}{2}}} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.6.4.3.1.2

Rewrite $11$ as $2$ plus $9$

$\frac{3 u^{2} + (2 + 9) u + 6}{t^{\frac{1}{2}}} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.6.4.3.1.3

Apply the distributive property.

$\frac{3 u^{2} + 2 u + 9 u + 6}{t^{\frac{1}{2}}} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.6.4.3.2

Factor out the greatest common factor from each group.

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Step 24.6.4.3.2.1

Group the first two terms and the last two terms.

$\frac{(3 u^{2} + 2 u) + 9 u + 6}{t^{\frac{1}{2}}} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.6.4.3.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{u (3 u + 2) + 3 (3 u + 2)}{t^{\frac{1}{2}}} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.6.4.3.3

Factor the polynomial by factoring out the greatest common factor, $3 u + 2$ .

$\frac{(3 u + 2) (u + 3)}{t^{\frac{1}{2}}} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.6.4.4

Replace all occurrences of $u$ with $t^{\frac{1}{2}}$ .

$\frac{(3 t^{\frac{1}{2}} + 2) (t^{\frac{1}{2}} + 3)}{t^{\frac{1}{2}}} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.7

To write $\frac{(3 t^{\frac{1}{2}} + 2) (t^{\frac{1}{2}} + 3)}{t^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{(3 t^{\frac{1}{2}} + 2) (t^{\frac{1}{2}} + 3)}{t^{\frac{1}{2}}} \cdot \frac{2}{2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.8

Write each expression with a common denominator of $2 t^{\frac{1}{2}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 24.8.1

Multiply $\frac{(3 t^{\frac{1}{2}} + 2) (t^{\frac{1}{2}} + 3)}{t^{\frac{1}{2}}}$ by $\frac{2}{2}$ .

$\frac{(3 t^{\frac{1}{2}} + 2) (t^{\frac{1}{2}} + 3) \cdot 2}{t^{\frac{1}{2}} \cdot 2} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.8.2

Move $2$ to the left of $t^{\frac{1}{2}}$ .

$\frac{(3 t^{\frac{1}{2}} + 2) (t^{\frac{1}{2}} + 3) \cdot 2}{2 t^{\frac{1}{2}}} + \frac{5}{2 t^{\frac{1}{2}}}$

Step 24.9

Combine the numerators over the common denominator.

$\frac{(3 t^{\frac{1}{2}} + 2) (t^{\frac{1}{2}} + 3) \cdot 2 + 5}{2 t^{\frac{1}{2}}}$

Step 24.10

Simplify the numerator.

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Step 24.10.1

Expand $(3 t^{\frac{1}{2}} + 2) (t^{\frac{1}{2}} + 3)$ using the FOIL Method.

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Step 24.10.1.1

Apply the distributive property.

$\frac{(3 t^{\frac{1}{2}} (t^{\frac{1}{2}} + 3) + 2 (t^{\frac{1}{2}} + 3)) \cdot 2 + 5}{2 t^{\frac{1}{2}}}$

Step 24.10.1.2

Apply the distributive property.

$\frac{(3 t^{\frac{1}{2}} t^{\frac{1}{2}} + 3 t^{\frac{1}{2}} \cdot 3 + 2 (t^{\frac{1}{2}} + 3)) \cdot 2 + 5}{2 t^{\frac{1}{2}}}$

Step 24.10.1.3

Apply the distributive property.

$\frac{(3 t^{\frac{1}{2}} t^{\frac{1}{2}} + 3 t^{\frac{1}{2}} \cdot 3 + 2 t^{\frac{1}{2}} + 2 \cdot 3) \cdot 2 + 5}{2 t^{\frac{1}{2}}}$

Step 24.10.2

Simplify and combine like terms.

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Step 24.10.2.1

Simplify each term.

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Step 24.10.2.1.1

Multiply $t^{\frac{1}{2}}$ by $t^{\frac{1}{2}}$ by adding the exponents.

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Step 24.10.2.1.1.1

Move $t^{\frac{1}{2}}$ .

$\frac{(3 (t^{\frac{1}{2}} t^{\frac{1}{2}}) + 3 t^{\frac{1}{2}} \cdot 3 + 2 t^{\frac{1}{2}} + 2 \cdot 3) \cdot 2 + 5}{2 t^{\frac{1}{2}}}$

Step 24.10.2.1.1.2

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

$\frac{(3 t^{\frac{1}{2} + \frac{1}{2}} + 3 t^{\frac{1}{2}} \cdot 3 + 2 t^{\frac{1}{2}} + 2 \cdot 3) \cdot 2 + 5}{2 t^{\frac{1}{2}}}$

Step 24.10.2.1.1.3

Combine the numerators over the common denominator.

$\frac{(3 t^{\frac{1 + 1}{2}} + 3 t^{\frac{1}{2}} \cdot 3 + 2 t^{\frac{1}{2}} + 2 \cdot 3) \cdot 2 + 5}{2 t^{\frac{1}{2}}}$

Step 24.10.2.1.1.4

Add $1$ and $1$ .

$\frac{(3 t^{\frac{2}{2}} + 3 t^{\frac{1}{2}} \cdot 3 + 2 t^{\frac{1}{2}} + 2 \cdot 3) \cdot 2 + 5}{2 t^{\frac{1}{2}}}$

Step 24.10.2.1.1.5

Divide $2$ by $2$ .

$\frac{(3 t^{1} + 3 t^{\frac{1}{2}} \cdot 3 + 2 t^{\frac{1}{2}} + 2 \cdot 3) \cdot 2 + 5}{2 t^{\frac{1}{2}}}$

Step 24.10.2.1.2

Simplify $3 t^{1}$ .

$\frac{(3 t + 3 t^{\frac{1}{2}} \cdot 3 + 2 t^{\frac{1}{2}} + 2 \cdot 3) \cdot 2 + 5}{2 t^{\frac{1}{2}}}$

Step 24.10.2.1.3

Multiply $3$ by $3$ .

$\frac{(3 t + 9 t^{\frac{1}{2}} + 2 t^{\frac{1}{2}} + 2 \cdot 3) \cdot 2 + 5}{2 t^{\frac{1}{2}}}$

Step 24.10.2.1.4

Multiply $2$ by $3$ .

$\frac{(3 t + 9 t^{\frac{1}{2}} + 2 t^{\frac{1}{2}} + 6) \cdot 2 + 5}{2 t^{\frac{1}{2}}}$

Step 24.10.2.2

Add $9 t^{\frac{1}{2}}$ and $2 t^{\frac{1}{2}}$ .

$\frac{(3 t + 11 t^{\frac{1}{2}} + 6) \cdot 2 + 5}{2 t^{\frac{1}{2}}}$

Step 24.10.3

Apply the distributive property.

$\frac{3 t \cdot 2 + 11 t^{\frac{1}{2}} \cdot 2 + 6 \cdot 2 + 5}{2 t^{\frac{1}{2}}}$

Step 24.10.4

Simplify.

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Step 24.10.4.1

Multiply $2$ by $3$ .

$\frac{6 t + 11 t^{\frac{1}{2}} \cdot 2 + 6 \cdot 2 + 5}{2 t^{\frac{1}{2}}}$

Step 24.10.4.2

Multiply $2$ by $11$ .

$\frac{6 t + 22 t^{\frac{1}{2}} + 6 \cdot 2 + 5}{2 t^{\frac{1}{2}}}$

Step 24.10.4.3

Multiply $6$ by $2$ .

$\frac{6 t + 22 t^{\frac{1}{2}} + 12 + 5}{2 t^{\frac{1}{2}}}$

Step 24.10.5

Add $12$ and $5$ .

$\frac{6 t + 22 t^{\frac{1}{2}} + 17}{2 t^{\frac{1}{2}}}$