Square roots and cube roots | Exponents

Working with numbers in exponential form

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.

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

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

Working with numbers in exponential form

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Calculations using numbers in exponential form