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# Surds

## 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}

$\sqrt{2^2 + 6^2} = 2\sqrt{10}$

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}

$\sqrt{18t^{2}} = 3\sqrt{2}t$

## 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.

$\sqrt{9 + k^2} \ne \sqrt{9} + \sqrt{k^2}$

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.

# Test yourself now

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Exercise 15.2

Write $$\sqrt{48}$$ in simplest surd form.

$\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$

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

$\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}$

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

$\sqrt{25 + 36} = \sqrt{61} = \text{7,81}$

Write $$\sqrt{175}$$ in simplest surd form.

$\sqrt{175} = \sqrt{25 \times 7} = 5\sqrt{7}$

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

$7p$

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

$\sqrt{9x^2 + 16}$

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

$\sqrt{80k^2} = \sqrt{16 \times 5 \times k^2} = 4k\sqrt{5} = 4\sqrt{5}k$

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

$\sqrt{b^2 + 3b^2} = \sqrt{4b^2} = 2b$