Simplify square root of a/4- square root of 4a^3

$\sqrt{\frac{a}{4}} - \sqrt{4 a^{3}}$

Step 1

Simplify each term.

Rewrite $\frac{a}{4}$ as ${(\frac{1}{2})}^{2} a$ .

Factor the perfect power $1^{2}$ out of $a$ .

$\sqrt{\frac{1^{2} a}{4}} - \sqrt{4 a^{3}}$

Factor the perfect power $2^{2}$ out of $4$ .

$\sqrt{\frac{1^{2} a}{2^{2} \cdot 1}} - \sqrt{4 a^{3}}$

Rearrange the fraction $\frac{1^{2} a}{2^{2} \cdot 1}$ .

$\sqrt{{(\frac{1}{2})}^{2} a} - \sqrt{4 a^{3}}$

Pull terms out from under the radical.

$\frac{1}{2} \sqrt{a} - \sqrt{4 a^{3}}$

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

$\frac{\sqrt{a}}{2} - \sqrt{4 a^{3}}$

Rewrite $4 a^{3}$ as ${(2 a)}^{2} a$ .

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

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

Factor out $a^{2}$ .

$\frac{\sqrt{a}}{2} - \sqrt{2^{2} (a^{2} a)}$

Rewrite $2^{2} a^{2}$ as ${(2 a)}^{2}$ .

$\frac{\sqrt{a}}{2} - \sqrt{{(2 a)}^{2} a}$

Pull terms out from under the radical.

$\frac{\sqrt{a}}{2} - (2 a \sqrt{a})$

Multiply $2$ by $- 1$ .

$\frac{\sqrt{a}}{2} - 2 a \sqrt{a}$

Step 2

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

$\frac{\sqrt{a}}{2} - 2 a \sqrt{a} \cdot \frac{2}{2}$

Step 3

Simplify terms.

Combine $- 2 a \sqrt{a}$ and $\frac{2}{2}$ .

$\frac{\sqrt{a}}{2} + \frac{- 2 a \sqrt{a} \cdot 2}{2}$

Combine the numerators over the common denominator.

$\frac{\sqrt{a} - 2 a \sqrt{a} \cdot 2}{2}$

Step 4

Simplify the numerator.

Factor $\sqrt{a}$ out of $\sqrt{a} - 2 a \sqrt{a} \cdot 2$ .

Multiply by $1$ .

$\frac{\sqrt{a} \cdot 1 - 2 a \sqrt{a} \cdot 2}{2}$

Factor $\sqrt{a}$ out of $- 2 a \sqrt{a} \cdot 2$ .

$\frac{\sqrt{a} \cdot 1 + \sqrt{a} (- 2 a \cdot 2)}{2}$

Factor $\sqrt{a}$ out of $\sqrt{a} \cdot 1 + \sqrt{a} (- 2 a \cdot 2)$ .

$\frac{\sqrt{a} (1 - 2 a \cdot 2)}{2}$

Multiply $2$ by $- 2$ .

$\frac{\sqrt{a} (1 - 4 a)}{2}$