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Square roots and cube roots

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Working with numbers in exponential form
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Calculations using numbers in exponential form

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.

Number $$\mathbf{1}$$ $$\mathbf{2}$$ $$\mathbf{3}$$ $$\mathbf{4}$$ $$\mathbf{5}$$ $$\mathbf{6}$$ $$\mathbf{7}$$ $$\mathbf{8}$$ $$\mathbf{9}$$ $$\mathbf{10}$$
Multiply by itself $$1 \times 1$$ $$2 \times 2$$ $$3 \times 3$$ $$4 \times 4$$ $$5 \times 5$$ $$6 \times 6$$ $$7 \times 7$$ $$8 \times 8$$ $$9 \times 9$$ $$10 \times 10$$
Exponential form $$1^{2}$$ $$2^{2}$$ $$3^{2}$$ $$4^{2}$$ $$5^{2}$$ $$6^{2}$$ $$7^{2}$$ $$8^{2}$$ $$9^{2}$$ $$10^{2}$$
Square $$1$$ $$4$$ $$9$$ $$16$$ $$25$$ $$36$$ $$49$$ $$64$$ $$81$$ $$100$$

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.

Number $$\mathbf{1}$$ $$\mathbf{4}$$ $$\mathbf{9}$$ $$\mathbf{16}$$ $$\mathbf{25}$$ $$\mathbf{36}$$ $$\mathbf{49}$$ $$\mathbf{64}$$ $$\mathbf{81}$$ $$\mathbf{100}$$ $$\mathbf{121}$$ $$\mathbf{144}$$
Square root $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$12$$
Check $$1 \times 1$$ $$2 \times 2$$ $$3 \times 3$$ $$4 \times 4$$ $$5 \times 5$$ $$6 \times 6$$ $$7 \times 7$$ $$8 \times 8$$ $$9 \times 9$$ $$10 \times 10$$ $$11 \times 11$$ $$12 \times 12$$

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.

Number $$\mathbf{1}$$ $$\mathbf{2}$$ $$\mathbf{3}$$ $$\mathbf{4}$$ $$\mathbf{5}$$ $$\mathbf{6}$$ $$\mathbf{7}$$ $$\mathbf{8}$$ $$\mathbf{9}$$ $$\mathbf{10}$$
Multiply by itself twice $$1 \times 1 \times 1$$ $$2 \times 2 \times 2$$ $$3 \times 3 \times 3$$ $$4 \times 4 \times 4$$ $$5 \times 5 \times 5$$ $$6 \times 6 \times 6$$ $$7 \times 7 \times 7$$ $$8 \times 8 \times 8$$ $$9 \times 9 \times 9$$ $$10 \times 10 \times 10$$
Exponential form $$1^{3}$$ $$2^{3}$$ $$3^{3}$$ $$4^{3}$$ $$5^{3}$$ $$6^{3}$$ $$7^{3}$$ $$8^{3}$$ $$9^{3}$$ $$10^{3}$$
Cube $$1$$ $$8$$ $$27$$ $$64$$ $$125$$ $$216$$ $$343$$ $$512$$ $$729$$ $$1000$$

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.

Number $$\mathbf{1}$$ $$\mathbf{8}$$ $$\mathbf{27}$$ $$\mathbf{64}$$ $$\mathbf{125}$$ $$\mathbf{216}$$ $$\mathbf{343}$$ $$\mathbf{512}$$ $$\mathbf{729}$$ $$\mathbf{1 000}$$
Cube root $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$
Check $$1 \times 1 \times 1$$ $$2 \times 2 \times 2$$ $$3 \times 3 \times 3$$ $$4 \times 4 \times 4$$ $$5 \times 5 \times 5$$ $$6 \times 6 \times 6$$ $$7 \times 7 \times 7$$ $$8 \times 8 \times 8$$ $$9 \times 9 \times 9$$ $$10 \times 10 \times 10$$

Sometimes you need to do some calculations to find the root.

Worked example 3.6: Finding square roots

Simplify and calculate the square root.

  1. \[\sqrt{4 \times 5 - 4}\]
  2. \[\sqrt{3 \times (10 + 2)}\]
  3. \[\sqrt{120 - 10 \times 2}\]
  4. \[\sqrt{33\ \times \ 3 + 1}\]

Do the calculations under the square root first.

  1. \[\sqrt{4 \times 5 - 4} = \sqrt{20 - 4} = \sqrt{16}\]
  2. \[\sqrt{3 \times (10 + 2)} = \sqrt{3 \times 12} = \sqrt{36}\]
  3. \[\sqrt{120 - 10 \times 2} = \sqrt{120 - 20} = \sqrt{100}\]
  4. \[\sqrt{33 \times 3 + 1} = \sqrt{99 + 1} = \sqrt{100}\]

Find the square root of the answer.

  1. \[\sqrt{16} = 4\]
  2. \[\sqrt{36} = 6\]
  3. \[\sqrt{100} = 10\]
  4. \[\sqrt{100} = 10\]

Worked example 3.7: Finding cube roots

Simplify and find the cube root.

  1. \[\sqrt[3]{200 + 16}\]
  2. \[\sqrt[3]{1000 - 271}\]
  3. \[\sqrt[3]{500 + 500}\]
  4. \[\sqrt[3]{13 + 26 + 25}\]

Do the calculations under the cube root first.

  1. \[\sqrt[3]{200 + 16} = \sqrt[3]{216}\]
  2. \[\sqrt[3]{1000 - 271} = \sqrt[3]{729}\]
  3. \[\sqrt[3]{500 + 500} = \sqrt[3]{1000}\]
  4. \[\sqrt[3]{13 + 26 + 25} = \sqrt[3]{64}\]

Find the cube root of the answer.

  1. \[\sqrt[3]{216} = 6\]
  2. \[\sqrt[3]{729} = 9\]
  3. \[\sqrt[3]{1000} = 10\]
  4. \[\sqrt[3]{64} = 4\]
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.

  1. \[\frac{81}{121}\]
  2. \[\frac{64}{81}\]
  3. \[\frac{49}{169}\]
  4. \[\frac{100}{225}\]

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.

  1. \[\frac{81}{121} = \frac{9 \times 9}{11 \times 11} = \frac{9}{11} \times \frac{9}{11}\]
  2. \[\frac{64}{81} = \frac{8 \times 8}{9 \times 9} = \frac{8}{9} \times \frac{8}{9}\]
  3. \[\frac{49}{169} = \frac{7 \times 7}{13 \times 13} = \frac{7}{13} \times \frac{7}{13}\]
  4. \[\frac{100}{225} = \frac{10 \times 10}{15 \times 15} = \frac{10}{15} \times \frac{10}{15}\]

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.

  1. \[\sqrt{\frac{16}{25}}\]
  2. \[\sqrt{\frac{9}{49}}\]
  3. \[\sqrt{\frac{81}{144}}\]
  4. \[\sqrt{\frac{400}{900}}\]

Use the rule you discovered in Question 1 to find these square roots.

  1. \[\sqrt{\frac{16}{25}} = \frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5}\]
  2. \[\sqrt{\frac{9}{49}} = \frac{\sqrt{9}}{\sqrt{49}} = \frac{3}{7}\]
  3. \[\sqrt{\frac{81}{144}} = \frac{\sqrt{81}}{\sqrt{144}} = \frac{9}{12}\]
  4. \[\sqrt{\frac{400}{900}} = \frac{\sqrt{400}}{\sqrt{900}} = \frac{20}{30} = \frac{2}{3}\]
  1. Use the fact that \(\text{0,01}\) can be written as \(\frac{1}{100}\) to calculate \(\sqrt{\text{0,01}}\).

  2. Use the fact that \(\text{0,49}\) can be written as \(\frac{49}{100}\) to calculate \(\sqrt{\text{0,49}}\).

  1. We know that \(\text{0,01}\) can be written as \(\frac{1}{100}\).
    So, \(\sqrt{\text{0,01}} = \sqrt{\frac{1}{100}} = \frac{\sqrt{1}}{\sqrt{100}} = \frac{1}{10} = \text{0,1}\).
  2. 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.

  1. \[\sqrt{\text{0,09}}\]
  2. \[\sqrt{\text{0,64}}\]
  3. \[\sqrt{\text{1,44}}\]
  4. \[\sqrt{\text{1,69}}\]

Find the square root of the number without a comma.

  1. \[\sqrt{09} = 3\]
  2. \[\sqrt{64} = 8\]
  3. \[\sqrt{144} = 12\]
  4. \[\sqrt{169} = 13\]

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.

  1. \(\text{0,09}\) 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: \(\text{0,3}\))

  2. \(\text{0,64}\) 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: \(\text{0,8}\))

  3. \(\text{1,44}\) 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: \(\text{1,2}\))

  4. \(\text{1,69}\) 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.

  1. \[\sqrt[3]{\frac{8}{27}}\]
  2. \[\sqrt[3]{\frac{343}{1000}}\]
  3. \[\sqrt[3]{\text{0,343}}\]
  4. \[\sqrt[3]{\frac{8000}{27000}}\]
  1. \[\sqrt[3]{\frac{8}{27}} = \frac{\sqrt[3]{8}}{\sqrt[3]{27}} = \frac{2}{3}\]
  2. \[\sqrt[3]{\frac{343}{1000}} = \frac{\sqrt[3]{343}}{\sqrt[3]{1000}} = \frac{7}{10} = \text{0,7}\]
  3. \[\sqrt[3]{\text{0,343}} = \text{0,7}\]

    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.

  4. \[\sqrt[3]{\frac{8000}{27000}} = \frac{\sqrt[3]{8000}}{\sqrt[3]{27000}} = \frac{20}{30} = \frac{2}{3}\] We could also simplify the fraction under the cube root before calculating:
    \[\sqrt[3]{\frac{8000}{27000}} = \sqrt[3]{\frac{8000 \div 1000}{27000 \div 1000}} = \sqrt[3]{\frac{8}{27}} = \frac{2}{3}\]
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