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Square roots and cube roots

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:
$\sqrt[3]{\frac{8000}{27000}} = \sqrt[3]{\frac{8000 \div 1000}{27000 \div 1000}} = \sqrt[3]{\frac{8}{27}} = \frac{2}{3}$