Mathematics » Exponents and Surds » Rational Exponents And Surds

# Simplest Surd Form

## Simplest Surd Form

We can sometimes simplify surds by writing the radicand as a product of factors that can be further simplified using $$\sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b}$$.

## Example

### Question

Write the following in simplest surd form: $$\sqrt{50}$$

### Write the radicand as a product of prime factors

\begin{align*} \sqrt{50} &= \sqrt{5 \times 5 \times 2} \\ &= \sqrt{5^2 \times 2} \end{align*}

### Simplify using $$\sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b}$$

\begin{align*} &= \sqrt{5^2} \times \sqrt{2} \\ &= 5 \times \sqrt{2} \\ &= 5\sqrt{2} \end{align*}

Sometimes a surd cannot be simplified. For example, $$\sqrt{6}, \sqrt[3]{30} \text{ and } \sqrt[4]{42}$$ are already in their simplest form.

## Example

### Question

Write the following in simplest surd form: $$\sqrt[3]{54}$$

### Write the radicand as a product of prime factors

\begin{align*} \sqrt[3]{54} &= \sqrt[3]{3 \times 3 \times 3 \times 2} \\ &= \sqrt[3]{3^3 \times 2} \end{align*}

### Simplify using $$\sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b}$$

\begin{align*} &= \sqrt[3]{3^3} \times \sqrt[3]{2} \\ &= 3 \times \sqrt[3]{2} \\ &= 3\sqrt[3]{2} \end{align*}

## Example

### Question

Simplify: $$\sqrt{\text{147}}+\sqrt{\text{108}}$$

### Write the radicands as a product of prime factors

\begin{align*} \sqrt{\text{147}} + \sqrt{\text{108}} &= \sqrt{49 \times 3} + \sqrt{36 \times 3} \\ &= \sqrt{7^2 \times 3} + \sqrt{6^2 \times 3} \end{align*}

### Simplify using $$\sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b}$$

\begin{align*} &= ( \sqrt{7^2} \times \sqrt{3} ) + ( \sqrt{6^2} \times \sqrt{3} )\\ &= ( 7 \times \sqrt{3} ) + ( 6 \times \sqrt{3} ) \\ &= 7\sqrt{3} + 6\sqrt{3} \end{align*}

### Simplify and write the final answer

$13 \sqrt{3}$

## Example

### Question

Simplify: $$( \sqrt{20} – \sqrt{5} )^2$$

### Factorise the radicands were possible

$( \sqrt{20} – \sqrt{5} )^2 = ( \sqrt{4 \times 5} – \sqrt{5} )^2$

### Simplify using $$\sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b}$$

\begin{align*} &= ( \sqrt{4} \times \sqrt{5} – \sqrt{5} )^2 \\ &= ( 2 \times \sqrt{5} – \sqrt{5} )^2 \\ &= ( 2\sqrt{5} – \sqrt{5} )^2 \end{align*}

### Simplify and write the final answer

\begin{align*} &= ( \sqrt{5} )^2 \\ &= 5 \end{align*}

## Example

### Question

Write in simplest surd form: $$\sqrt{75} \times \sqrt[3]{(48)^{-1}}$$

### Factorise the radicands were possible

\begin{align*} \sqrt{75} \times \sqrt[3]{(48)^{-1}} &= \sqrt{25 \times 3} \times \sqrt[3]{\cfrac{1}{48}} \\ &= \sqrt{25 \times 3} \times \cfrac{1}{\sqrt[3]{8 \times 6}} \end{align*}

### Simplify using $$\sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b}$$

\begin{align*} &= \sqrt{25} \times \sqrt{3} \times \cfrac{1}{\sqrt[3]{8} \times \sqrt[3]{6}} \\ &= 5 \times \sqrt{3} \times \cfrac{1}{2 \times \sqrt[3]{6}} \end{align*}

### Simplify and write the final answer

\begin{align*} &= 5\sqrt{3} \times \cfrac{1}{2\sqrt[3]{6}} \\ &= \cfrac{5\sqrt{3}}{2\sqrt[3]{6}} \end{align*}