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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 \(4\):

\[\begin{align} \sqrt{40} &= \sqrt{4 \times 10} \\ &= 2\sqrt{10} \end{align}\]

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.

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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.

\[\sqrt{b^2 + 3b^2} = \sqrt{4b^2} = 2b\]