Simplify p^(1/2)(p^(1/2)+3p^(3/2))

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

Step 1

Apply the distributive property.

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

Step 2

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

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $1$ .

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

Divide $2$ by $2$ .

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

Step 3

Simplify $p^{1}$ .

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

Step 4

Rewrite using the commutative property of multiplication.

$p + 3 p^{\frac{1}{2}} p^{\frac{3}{2}}$

Step 5

Multiply $p^{\frac{1}{2}}$ by $p^{\frac{3}{2}}$ by adding the exponents.

Move $p^{\frac{3}{2}}$ .

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

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

$p + 3 p^{\frac{3}{2} + \frac{1}{2}}$

Combine the numerators over the common denominator.

$p + 3 p^{\frac{3 + 1}{2}}$

Add $3$ and $1$ .

$p + 3 p^{\frac{4}{2}}$

Divide $4$ by $2$ .

$p + 3 p^{2}$