Write \(\sqrt{48}\) in simplest surd form.

## 15.3 Surds

So far, we have looked at the square root of perfect squares. How do we handle numbers like \(\sqrt{5},
\sqrt{11}\) or \(\sqrt{20}\), where the radicands are not perfect squares? A square root that cannot be
simplified is called a **surd**. If we cannot simplify to remove the square root, then we say that
the answer is in surd form. If a question states that the answer must be given in simplest surd form, we need to
make sure that the radicand is in its simplest form.

- surd
- a square root that cannot be simplified

## Worked Example 15.4: Simplifying expressions with square root signs

Simplify \(\sqrt{2^2 + 6^2}\). Give your answer in simplest surd form.

### Simplify the expression underneath the square root sign.

First we need to simplify the expression underneath the square root sign. Remember always to apply the correct order of operations.

\[\begin{align} \sqrt{2^2 + 6^2} &= \sqrt{4 + 36} \\ &= \sqrt{40} \end{align}\]Notice that \(40\) is not a perfect square.

### Write down the factors of the radicand.

We write down the factors of \(40\) to see if we can simplify the expression:

\[40 = 4 \times 10\]We can simplify the expression by taking the square root of \(4\):

\[\begin{align} \sqrt{40} &= \sqrt{4 \times 10} \\ &= 2\sqrt{10} \end{align}\]### Write the final answer.

\[\sqrt{2^2 + 6^2} = 2\sqrt{10}\]Can the expression \(\sqrt{3} + \sqrt{7}\) be simplified? No, these two terms are already in simplest surd form.

## Worked Example 15.5: Simplifying algebraic expressions with square root signs

Write \(\sqrt{18t^{2}}\) in its simplest form.

### Evaluate the square root.

The radicand consists of two parts: a coefficient and a variable of power \(2\).

- The coefficient is not a perfect square: \(18 = 3^2 \times 2\).
- The algebraic part is a perfect square: \(t \times t = t^{2}\).

### Simplify the expression.

\[\sqrt{18t^{2}} = \sqrt{3^2 \times 2 \times t^2}\]So,

\[\begin{align} \sqrt{18t^{2}} &= 3t\sqrt{2} \\ &= 3\sqrt{2}t \end{align}\]### Write the final answer.

\[\sqrt{18t^{2}} = 3\sqrt{2}t\]## Worked Example 15.6: Simplifying algebraic expressions with square root signs

Sindisiwe simplified the expression \(\sqrt{9 + k^2}\) as shown below:

Sindisiwe’s calculation: | Reason: |
---|---|

\(\sqrt{9 + k^2}\) | |

\(= \sqrt{9} + \sqrt{k^2}\) | \(9\) is perfect square and \(k^2\) is perfect square |

\(= 3 + k\) |

Do you agree with the method used to calculate the answer? If not, provide the correct solution.

### Check each step of the given solution.

Sindisiwe is correct that both \(9\) and \(k^2\) are perfect squares, but these two unlike terms are underneath a square root sign and they cannot be split up into two square roots.

\[\sqrt{9 + k^2} \ne \sqrt{9} + \sqrt{k^2}\]### Write the final answer.

We cannot simplify the expression. The expression \(\sqrt{9 + k^2}\) is already in its simplest form.

If a question states that we must give the answer correct to a certain number of decimal numbers, we can use the \(\surd\) sign on a calculator to determine the answer. Make sure you know where this button is on your calculator.

Simplify \(\sqrt{45}\). Give your answer in simplest surd form.

Evaluate \(\sqrt{25 + 36}\). Give your answer correct to two decimal places.

Write \(\sqrt{175}\) in simplest surd form.

Evaluate \(\sqrt{49p^2}\). Give your answer correct to two decimal places.

Simplify \(\sqrt{9x^2 + 16}\). Give your answer in simplest surd form.

Write \(\sqrt{80k^2}\) in simplest surd form.

Simplify \(\sqrt{b^2 + 3b^2}\). Give your answer in simplest surd form.