Evaluate \(\sqrt{25 + 144}\).

## 15.2 Perfect squares and square roots

### Perfect squares

Remember that the **square of a number** is a number multiplied by itself.

For example:

- The square of \(3\) is \(9\) because \(3 \times 3 = 3^2 = 9\). We say \(3\) squared equals \(9\).
- The square of \(7\) is \(49\) because \(7 \times 7 = 7^2 = 49\). We say \(7\) squared equals \(49\).

We say that \(9\) and \(49\) are perfect squares.

Notice that \(s \times s = s^2\) is the formula for the area of a square with each side equal to \(s\).

Area of square \(= s \times s = s^2\)

Copy and complete the table. The first three rows have been completed as examples.

Number | Square of number in exponential form | Perfect square |
---|---|---|

\(0\) | \(0^{2}\) | \(0\) |

\(1\) | \(1^{2}\) | \(1\) |

\(2\) | \(2^{2}\) | \(4\) |

\(3\) | ||

\(4\) | ||

\(5\) | ||

\(6\) | ||

\(7\) | ||

\(8\) | ||

\(9\) | ||

\(10\) | ||

\(11\) | ||

\(12\) |

### Square roots

To find the square root of a number, we ask the question: Which number was multiplied by itself to get the square?

Remember:

- The square root of \(16\) is \(4\) because \(4 \times 4 = 16\). We write this as \(\sqrt{16} = 4\).
- The square root of \(121\) is \(11\) because \(11 \times 11 = 121\). We write this as \(\sqrt{121} = 11\).

The number underneath the square root sign is called the **radicand**.

- radicand
- the number underneath the square root sign

Copy and complete the table. The first three rows have been completed as examples.

Square root of perfect square | Square root |
---|---|

\(\sqrt{0}\) | \(0\) |

\(\sqrt{1}\) | \(1\) |

\(\sqrt{4}\) | \(2\) |

\(\sqrt{9}\) | |

\(\sqrt{16}\) | |

\(\sqrt{25}\) | |

\(\sqrt{36}\) | |

\(\sqrt{49}\) | |

\(\sqrt{64}\) | |

\(\sqrt{81}\) | |

\(\sqrt{100}\) | |

\(\sqrt{121}\) | |

\(\sqrt{144}\) |

## Worked Example 15.1: Simplifying expressions with square root signs

Write \(\sqrt{5^{2} + 12^2}\) in its simplest 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.

\[\begin{align} \sqrt{5^{2} + 12^2} &= \sqrt{25 + 144} \\ &= \sqrt{169} \end{align}\]### Determine the square root.

We know that

\[\begin{align} 13 \times 13 &= 13^{2} \\ &= 169 \\ \therefore \sqrt{169} &= 13 \end{align}\]## Worked Example 15.2: Simplifying expressions with square root signs

Simplify \(\sqrt{16} + \sqrt{9}\).

### Determine the square root for each term.

We first find the square root of each term and then we add the two values together.

We know that \(\sqrt{16} = 4\) and \(\sqrt{9} = 3\) So,

\[\begin{align} \sqrt{16} + \sqrt{9} &= 4 + 3 \\ &= 7 \end{align}\]We

\[\begin{align} \sqrt{16} + \sqrt{9} &= \sqrt{16 + 9} \\ &= \sqrt{25} \\ &= 5 \end{align}\]cannotcombine the two square roots by adding the two radicands together. If we use this incorrect method, we get the wrong answer. Incorrect method:We can only combine square root signs if there is a multiplication or division sign between them.

### Write the final answer.

\[\sqrt{16} + \sqrt{9} = 7\]## Worked Example 15.3: Simplifying expressions with square root signs

Simplify \(\sqrt{25} \times \sqrt{4}\).

### Method 1: Evaluate the square roots and then find product.

\[\begin{align} \sqrt{25} \times \sqrt{4} &= 5 \times 2 \\ &= 10 \end{align}\]### Method 2: Combine the square roots and then determine the final answer.

There is a multiplication sign between the two square roots, so we can combine the two square roots and simplify the combined expression:

\[\begin{align} \sqrt{25} \times \sqrt{4} &= \sqrt{25 \times 4} \\ &= \sqrt{100} \\ &= 10 \end{align}\]Notice that Method 1 and Method 2 give the same answer.

### Write the final answer.

\[\sqrt{25} \times \sqrt{4} = 10\]Simplify \(\sqrt{100} + \sqrt{49}\).

Write \(\sqrt{36} − \sqrt{9}\) in its simplest form.

Evaluate \(2\sqrt{64}\).

Simplify \(\sqrt{16} − 4\).

Write \(\sqrt{81} + \sqrt{1}\) in its simplest form.

Simplify \(5\sqrt{100}\).

\(5 \times 10 = 50\).

Evaluate \(3\sqrt{9} − 3\).