3.3 Square roots and cube roots
Squares and square roots
To square a number is to multiply it by itself. The square of \(8\) is \(64\) because \(8 \times 8\) equals
\(64\).
We write \(8 \times 8\) as \(8^{2}\) in exponential form.
We read \(8^{2}\) as eight squared. The number \(64\) is a square number.
 square number
 the product of a number multiplied by itself
To calculate the area of a square (equal sides), we multiply the side length by itself. If the
area of a square is \(64 \text{ cm}^{2}\) (square centimetres), then the sides of that square are \(8 \text{
cm}\).
Look at the first ten positive square numbers.
Can you see a pattern in the last row in the table above?
\[4  1 = 3\]
\[9  4 = 5\]
\[16  9 = 7\]
\[25  16 = 9\]
\[36  25 =\text{ ?}\]
The difference between consecutive square numbers is always an odd number.
To find the square root of a number, we ask the question: Which number was multiplied by itself to get the
square?
The square root of \(16\) is \(4\) because \(4 \times 4 = 16\).
The question: “Which number was multiplied by itself to get \(16\)?” is written mathematically as
\(\sqrt{16}\).
The answer to this question is written as \(\sqrt{16} = 4\).
Look at the first twelve square numbers and their square roots.
Cubes and cube roots
To cube a number is to multiply it by itself and then by itself again. The cube of \(3\) is \(27\) because \(3
\times 3 \times 3\) equals \(27\).
We write \(3 \times 3 \times 3\) as \(3^{3}\) in exponential form.
We read \(3^{3}\) as three cubed. The number \(27\) is a cube number.
 cube number
 the product of a number multiplied by itself and then by itself again
To calculate the volume of a cube (equal sides), we multiply the side length by itself twice. If
the volume of a cube is \(27 \text{ cm}^{3}\) (cubic centimetres), then the sides of that cube are \(3 \text{
cm}\).
Look at the first ten positive cube numbers.
To find the cube root of a number, we ask the question: Which number was multiplied by itself and again by
itself to get the cube?
The cube root of \(64\) is \(4\) because \(4 \times 4 \times 4 = 64\).
The question: “Which number was multiplied by itself and again by itself (or cubed) to get \(64\)?”
is written mathematically as \(\sqrt[3]{64}\).
The answer to this question is written as \(\sqrt[3]{64} = 4\).
Look at the first ten cube numbers and their cube roots.
Sometimes you need to do some calculations to find the root.
Worked example 3.6: Finding square roots
Simplify and calculate the square root.

\[\sqrt{4 \times 5  4}\]

\[\sqrt{3 \times (10 + 2)}\]

\[\sqrt{120  10 \times 2}\]

\[\sqrt{33\ \times \ 3 + 1}\]
Do the calculations under the square root first.

\[\sqrt{4 \times 5  4} = \sqrt{20  4} = \sqrt{16}\]

\[\sqrt{3 \times (10 + 2)} = \sqrt{3 \times 12} = \sqrt{36}\]

\[\sqrt{120  10 \times 2} = \sqrt{120  20} = \sqrt{100}\]

\[\sqrt{33 \times 3 + 1} = \sqrt{99 + 1} = \sqrt{100}\]
Find the square root of the answer.

\[\sqrt{16} = 4\]

\[\sqrt{36} = 6\]

\[\sqrt{100} = 10\]

\[\sqrt{100} = 10\]
Worked example 3.7: Finding cube roots
Simplify and find the cube root.

\[\sqrt[3]{200 + 16}\]

\[\sqrt[3]{1000  271}\]

\[\sqrt[3]{500 + 500}\]

\[\sqrt[3]{13 + 26 + 25}\]
Do the calculations under the cube root first.

\[\sqrt[3]{200 + 16} = \sqrt[3]{216}\]

\[\sqrt[3]{1000  271} = \sqrt[3]{729}\]

\[\sqrt[3]{500 + 500} = \sqrt[3]{1000}\]

\[\sqrt[3]{13 + 26 + 25} = \sqrt[3]{64}\]
Find the cube root of the answer.

\[\sqrt[3]{216} = 6\]

\[\sqrt[3]{729} = 9\]

\[\sqrt[3]{1000} = 10\]

\[\sqrt[3]{64} = 4\]
Exercise 3.8: Finding square roots of fractions and
decimals
Write the fraction as a product of two equal factors to work out the square root.

\[\frac{81}{121}\]

\[\frac{64}{81}\]

\[\frac{49}{169}\]

\[\frac{100}{225}\]
We know that to find a square root is to find the number which when multiplied by itself gives the
square. In this example, we are looking for a product of two fractions that are the same.

\[\frac{81}{121} = \frac{9 \times 9}{11 \times 11} = \frac{9}{11} \times \frac{9}{11}\]

\[\frac{64}{81} = \frac{8 \times 8}{9 \times 9} = \frac{8}{9} \times \frac{8}{9}\]

\[\frac{49}{169} = \frac{7 \times 7}{13 \times 13} = \frac{7}{13} \times \frac{7}{13}\]

\[\frac{100}{225} = \frac{10 \times 10}{15 \times 15} = \frac{10}{15} \times \frac{10}{15}\]
Do you see the pattern? To find the square root of a fraction, find the square root of
the numerator and the denominator. So, \(\sqrt{\frac{4}{9}} = \frac{\sqrt{4}}{\sqrt{9}} =
\frac{2}{3}\).
Determine the following.

\[\sqrt{\frac{16}{25}}\]

\[\sqrt{\frac{9}{49}}\]

\[\sqrt{\frac{81}{144}}\]

\[\sqrt{\frac{400}{900}}\]
Use the rule you discovered in Question 1 to find these square roots.

\[\sqrt{\frac{16}{25}} = \frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5}\]

\[\sqrt{\frac{9}{49}} = \frac{\sqrt{9}}{\sqrt{49}} = \frac{3}{7}\]

\[\sqrt{\frac{81}{144}} = \frac{\sqrt{81}}{\sqrt{144}} = \frac{9}{12}\]

\[\sqrt{\frac{400}{900}} = \frac{\sqrt{400}}{\sqrt{900}} = \frac{20}{30} = \frac{2}{3}\]
 Use the fact that \(\text{0,01}\) can be written as \(\frac{1}{100}\) to calculate \(\sqrt{\text{0,01}}\).

Use the fact that \(\text{0,49}\) can be written as \(\frac{49}{100}\) to calculate \(\sqrt{\text{0,49}}\).
 We know that \(\text{0,01}\) can be written as \(\frac{1}{100}\).
So, \(\sqrt{\text{0,01}} = \sqrt{\frac{1}{100}} = \frac{\sqrt{1}}{\sqrt{100}} = \frac{1}{10} = \text{0,1}\).
 We know that \(\text{0,49}\) can be written as \(\frac{49}{100}\).
So, \(\sqrt{\text{0,49}} = \sqrt{\frac{49}{100}} = \frac{\sqrt{49}}{\sqrt{100}} = \frac{7}{10} = \text{0,7}\).
Do you see the pattern? To find the square root of a decimal number:
Step 1: Find the square root of the number without the comma.
Step 2: Check the number of digits to the right of the comma in the given decimal number.
Move the comma half the number of places in the answer.
For example, \(\sqrt{\text{0,36}}\).
Step 1: \(\sqrt{36} = 6\)
Step 2: \(\text{0,36}\) has two digits after the comma. The answer must have only one digit.
So, \(\sqrt{\text{0,36}} = \text{0,6}\).
Worked example 3.8: Finding square
roots of fractions and decimals
Calculate the following.

\[\sqrt{\text{0,09}}\]

\[\sqrt{\text{0,64}}\]

\[\sqrt{\text{1,44}}\]

\[\sqrt{\text{1,69}}\]
Find the square root of the number without a
comma.

\[\sqrt{09} = 3\]

\[\sqrt{64} = 8\]

\[\sqrt{144} = 12\]

\[\sqrt{169} = 13\]
Check the number of digits to the right of the comma in the given decimal number. Move the comma half the
number of places in the answer.

\(\text{0,09}\) has two digits after the comma, so the answer has only one digit.
\(\sqrt{\text{0,09}} = \text{0,3}\) (\(\sqrt{9} = 3\) and only one place after the comma: \(\text{0,3}\))

\(\text{0,64}\) has two digits after the comma, so the answer has only one digit.
\(\sqrt{\text{0,64}} = \text{0,8}\) (\(\sqrt{64} = 8\) and only one place after the comma: \(\text{0,8}\))

\(\text{1,44}\) has two digits after the comma, so the answer has only one digit.
\(\sqrt{\text{1,44}} = \text{1,2}\) (\(\sqrt{144} = 12\) and only one place after the comma: \(\text{1,2}\))

\(\text{1,69}\) has two digits after the comma, so the answer has only one digit.
\(\sqrt{\text{1,69}} = \text{1,3}\) (\(\sqrt{169} = 13\) and only one place after the comma: \(\text{1,3}\))
Exercise 3.9: Finding cube roots of fractions and decimals
Find the cube roots of the following fractions and decimals.

\[\sqrt[3]{\frac{8}{27}}\]

\[\sqrt[3]{\frac{343}{1000}}\]

\[\sqrt[3]{\text{0,343}}\]

\[\sqrt[3]{\frac{8000}{27000}}\]

\[\sqrt[3]{\frac{8}{27}} = \frac{\sqrt[3]{8}}{\sqrt[3]{27}} = \frac{2}{3}\]

\[\sqrt[3]{\frac{343}{1000}} = \frac{\sqrt[3]{343}}{\sqrt[3]{1000}} = \frac{7}{10} = \text{0,7}\]

\[\sqrt[3]{\text{0,343}} = \text{0,7}\]
Check that the answer works: \(\text{0,7} \times \ \text{0,7} \times \text{0,7} = \text{0,49} \times \text{0,7} =
\text{0,343}\). Can you see what happened to the number of digits after the comma? The number under the
cube root had \(3\) digits, but the answer has \(1\) digit.
 \[\sqrt[3]{\frac{8000}{27000}} = \frac{\sqrt[3]{8000}}{\sqrt[3]{27000}} = \frac{20}{30} =
\frac{2}{3}\]
We could also simplify the fraction under the cube root before calculating:
\[\sqrt[3]{\frac{8000}{27000}} = \sqrt[3]{\frac{8000 \div 1000}{27000 \div 1000}} =
\sqrt[3]{\frac{8}{27}} = \frac{2}{3}\]