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Perfect squares and square roots

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
\(0\) \(0^{2}\) \(0\)
\(1\) \(1^{2}\) \(1\)
\(2\) \(2^{2}\) \(4\)
\(3\)    
\(4\)    
\(5\)    
\(6\)    
\(7\)    
\(8\)    
\(9\)    
\(10\)    
\(11\)    
\(12\)    
temp text

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 cannot combine the two square roots by adding the two radicands together. If we use this incorrect method, we get the wrong answer. Incorrect method:

\[\begin{align} \sqrt{16} + \sqrt{9} &= \sqrt{16 + 9} \\ &= \sqrt{25} \\ &= 5 \end{align}\]

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.

\[\sqrt{25} \times \sqrt{4} = 10\]
Exercise 15.1

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

\[\sqrt{169} = 13\]

Simplify \(\sqrt{100} + \sqrt{49}\).

\[10 + 7 = 17\]

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

\[6 − 3 = 3\]

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

\[2 \times 8 = 16\]

Simplify \(\sqrt{16} − 4\).

\[4 − 4 = 0\]

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

\[9 + 1 = 10\]

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

\(5 \times 10 = 50\).

Evaluate \(3\sqrt{9} − 3\).

\[3(3) − 3 = 9 − 3 = 6\]