Basic Math Examples

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

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Rewrite using the commutative property of multiplication.

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

Step 1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.1

Factor $2$ out of $2 \sqrt{t^{2} t^{3}}$ .

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Rewrite the expression.

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

Step 1.3

Simplify the denominator.

Tap for more steps...

Step 1.3.1

Multiply $t^{2}$ by $t^{3}$ by adding the exponents.

Tap for more steps...

Step 1.3.1.1

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

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

Step 1.3.1.2

Add $2$ and $3$ .

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

Step 1.3.2

Rewrite $t^{5}$ as ${(t^{2})}^{2} t$ .

Tap for more steps...

Step 1.3.2.1

Factor out $t^{4}$ .

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

Step 1.3.2.2

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

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

Step 1.3.3

Pull terms out from under the radical.

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

Step 1.4

Cancel the common factor of $t$ .

Tap for more steps...

Step 1.4.1

Factor $t$ out of $t^{2} \sqrt{t}$ .

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

Step 1.4.2

Cancel the common factor.

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

Step 1.4.3

Rewrite the expression.

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

Step 1.5

Cancel the common factor of $t^{3}$ and $t$ .

Tap for more steps...

Step 1.5.1

Factor $t$ out of $t^{3}$ .

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

Step 1.5.2

Cancel the common factors.

Tap for more steps...

Step 1.5.2.1

Factor $t$ out of $t \sqrt{t}$ .

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

Step 1.5.2.2

Cancel the common factor.

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

Step 1.5.2.3

Rewrite the expression.

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

Step 1.6

Multiply $\frac{t^{2}}{\sqrt{t}}$ by $\frac{\sqrt{t}}{\sqrt{t}}$ .

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

Step 1.7

Combine and simplify the denominator.

Tap for more steps...

Step 1.7.1

Multiply $\frac{t^{2}}{\sqrt{t}}$ by $\frac{\sqrt{t}}{\sqrt{t}}$ .

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

Step 1.7.2

Raise $\sqrt{t}$ to the power of $1$ .

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

Step 1.7.3

Raise $\sqrt{t}$ to the power of $1$ .

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

Step 1.7.4

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

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

Step 1.7.5

Add $1$ and $1$ .

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

Step 1.7.6

Rewrite ${\sqrt{t}}^{2}$ as $t$ .

Tap for more steps...

Step 1.7.6.1

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

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

Step 1.7.6.2

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

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

Step 1.7.6.3

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

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

Step 1.7.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.7.6.4.1

Cancel the common factor.

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

Step 1.7.6.4.2

Rewrite the expression.

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

Step 1.7.6.5

Simplify.

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

Step 1.8

Cancel the common factor of $t^{2}$ and $t$ .

Tap for more steps...

Step 1.8.1

Factor $t$ out of $t^{2} \sqrt{t}$ .

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

Step 1.8.2

Cancel the common factors.

Tap for more steps...

Step 1.8.2.1

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

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

Step 1.8.2.2

Factor $t$ out of $t^{1}$ .

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

Step 1.8.2.3

Cancel the common factor.

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

Step 1.8.2.4

Rewrite the expression.

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

Step 1.8.2.5

Divide $t \sqrt{t}$ by $1$ .

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

Step 1.9

Rewrite using the commutative property of multiplication.

$t \sqrt{t} + 3 \frac{t^{2}}{2 \sqrt{t^{2} + t^{3}}} t^{2}$

Step 1.10

Simplify the denominator.

Tap for more steps...

Step 1.10.1

Factor $t^{2}$ out of $t^{2} + t^{3}$ .

Tap for more steps...

Step 1.10.1.1

Multiply by $1$ .

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

Step 1.10.1.2

Factor $t^{2}$ out of $t^{3}$ .

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

Step 1.10.1.3

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

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

Step 1.10.2

Rewrite $1$ as $1^{2}$ .

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

Step 1.10.3

Pull terms out from under the radical.

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

Step 1.10.4

One to any power is one.

$t \sqrt{t} + 3 \frac{t^{2}}{2 t \sqrt{1 + t}} t^{2}$

Step 1.11

Cancel the common factor of $t^{2}$ and $t$ .

Tap for more steps...

Step 1.11.1

Factor $t$ out of $t^{2}$ .

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

Step 1.11.2

Cancel the common factors.

Tap for more steps...

Step 1.11.2.1

Factor $t$ out of $2 t \sqrt{1 + t}$ .

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

Step 1.11.2.2

Cancel the common factor.

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

Step 1.11.2.3

Rewrite the expression.

$t \sqrt{t} + 3 \frac{t}{2 \sqrt{1 + t}} t^{2}$

Step 1.12

Multiply $\frac{t}{2 \sqrt{1 + t}}$ by $\frac{\sqrt{1 + t}}{\sqrt{1 + t}}$ .

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

Step 1.13

Combine and simplify the denominator.

Tap for more steps...

Step 1.13.1

Multiply $\frac{t}{2 \sqrt{1 + t}}$ by $\frac{\sqrt{1 + t}}{\sqrt{1 + t}}$ .

$t \sqrt{t} + 3 \frac{t \sqrt{1 + t}}{2 \sqrt{1 + t} \sqrt{1 + t}} t^{2}$

Step 1.13.2

Move $\sqrt{1 + t}$ .

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

Step 1.13.3

Raise $\sqrt{1 + t}$ to the power of $1$ .

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

Step 1.13.4

Raise $\sqrt{1 + t}$ to the power of $1$ .

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

Step 1.13.5

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

$t \sqrt{t} + 3 \frac{t \sqrt{1 + t}}{2 {\sqrt{1 + t}}^{1 + 1}} t^{2}$

Step 1.13.6

Add $1$ and $1$ .

$t \sqrt{t} + 3 \frac{t \sqrt{1 + t}}{2 {\sqrt{1 + t}}^{2}} t^{2}$

Step 1.13.7

Rewrite ${\sqrt{1 + t}}^{2}$ as $1 + t$ .

Tap for more steps...

Step 1.13.7.1

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

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

Step 1.13.7.2

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

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

Step 1.13.7.3

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

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

Step 1.13.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.13.7.4.1

Cancel the common factor.

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

Step 1.13.7.4.2

Rewrite the expression.

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

Step 1.13.7.5

Simplify.

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

Step 1.14

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

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

Step 1.15

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

Tap for more steps...

Step 1.15.1

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

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

Step 1.15.2

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

Tap for more steps...

Step 1.15.2.1

Move $t^{2}$ .

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

Step 1.15.2.2

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

Tap for more steps...

Step 1.15.2.2.1

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

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

Step 1.15.2.2.2

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

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

Step 1.15.2.3

Add $2$ and $1$ .

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

Step 2

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

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

Step 3

Simplify terms.

Tap for more steps...

Step 3.1

Combine $t \sqrt{t}$ and $\frac{2 (1 + t)}{2 (1 + t)}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

Tap for more steps...

Step 4.1

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

Tap for more steps...

Step 4.1.1

Factor $t$ out of $t \sqrt{t} (2 (1 + t))$ .

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

Step 4.1.2

Factor $t$ out of $3 t^{3} \sqrt{1 + t}$ .

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

Step 4.1.3

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Rewrite using the commutative property of multiplication.

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

Step 4.5

Apply the distributive property.

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

Step 4.6

Multiply $2$ by $1$ .

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

Step 4.7

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

Tap for more steps...

Step 4.7.1

Move $t$ .

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

Step 4.7.2

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

Tap for more steps...

Step 4.7.2.1

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

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

Step 4.7.2.2

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

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

Step 4.7.3

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

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

Step 4.7.4

Combine the numerators over the common denominator.

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

Step 4.7.5

Add $2$ and $1$ .

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

Step 4.8

Reorder terms.

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

Step 4.9

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

Tap for more steps...

Step 4.9.1

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

Tap for more steps...

Step 4.9.1.1

Reorder $1$ and $t$ .

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

Step 4.9.1.2

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

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

Step 4.9.1.3

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

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

Step 4.9.1.4

Factor $t^{\frac{1}{2}}$ out of $2 t^{\frac{3}{2}}$ .

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

Step 4.9.1.5

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

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

Step 4.9.1.6

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

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

Step 4.9.2

Divide $2$ by $2$ .

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

Step 4.9.3

Simplify.

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

Step 4.9.4

Reorder terms.

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

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

Step 4.10

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

Tap for more steps...

Step 4.10.1

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

Tap for more steps...

Step 4.10.1.1

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

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

Step 4.10.1.2

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

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

Step 4.10.2

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

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

Step 4.10.3

Combine the numerators over the common denominator.

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

Step 4.10.4

Add $2$ and $1$ .

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