3.1.2 Squares, Square Roots, Cube and Cube Roots (II) - Mathematics

3.1.2 Squares, Square Roots, Cube and Cube Roots

(C) Square Roots

1. The square root of a positive number is a number multiplied by itself whose product is equal to the given number.

Example:

\begin{matrix} (a) \sqrt{169} = \sqrt{13 \times 13} = 13 (b) \sqrt{\frac{25}{64}} = \sqrt{\frac{5 \times 5}{8 \times 8}} = \frac{5}{8} (c) \sqrt{\frac{72}{98}} = \sqrt{\frac{7236}{9849}} = \sqrt{\frac{6 \times 6}{7 \times 7}} = \frac{6}{7} (d) \sqrt{3 \frac{1}{16}} = \sqrt{\frac{49}{16}} = \frac{7}{4} = 1 \frac{3}{4} (e) \sqrt{1.44} = \sqrt{1 \frac{4411}{10025}} = \sqrt{\frac{36}{25}} = \frac{6}{5} = 1 \frac{1}{5} \end{matrix}

$\begin{array}{l} (a) \sqrt{169} = \sqrt{13 \times 13} = 13 \\ (b) \sqrt{\frac{25}{64}} = \sqrt{\frac{5 \times 5}{8 \times 8}} = \frac{5}{8} \\ (c) \sqrt{\frac{72}{98}} = \sqrt{\frac{7236}{9849}} = \sqrt{\frac{6 \times 6}{7 \times 7}} = \frac{6}{7} \\ (d) \sqrt{3 \frac{1}{16}} = \sqrt{\frac{49}{16}} = \frac{7}{4} = 1 \frac{3}{4} \\ (e) \sqrt{1.44} = \sqrt{1 \frac{4411}{10025}} = \sqrt{\frac{36}{25}} = \frac{6}{5} = 1 \frac{1}{5} \end{array}$

(D) Cubes

1. The cube of a number is obtained when that number is multiplied by itself twice.

Example:

The cube of 3 is written as

3³ = 3 × 3 × 3

= 27

2. The cube of a negative number is negative.

Example:

(–2)³ = (–2) × (–2) × (–2)

= –8

3. The cube of zero is zero. The cube of one is one, 1³ = 1.

(E) Cube Roots

1. The cube root of a number is a number which, when multiplied by itself twice, produces the particular number.

" \sqrt[3]{} "

$" \sqrt[3]{} "$ is the symbol for cube root.

Example:

\begin{matrix} \sqrt[3]{64} = \sqrt[3]{4 \times 4 \times 4} = 4 \end{matrix}

$\begin{array}{l} \sqrt[3]{64} = \sqrt[3]{4 \times 4 \times 4} \\ = 4 \end{array}$

\sqrt[3]{64}

$\sqrt[3]{64}$ is read as ‘cube root of sixty-four’.

2. The cube root of a positive number is positive.

Example:

\begin{matrix} \sqrt[3]{125} = \sqrt[3]{5 \times 5 \times 5} = 5 \end{matrix}

$\begin{array}{l} \sqrt[3]{125} = \sqrt[3]{5 \times 5 \times 5} \\ = 5 \end{array}$

3. The cube root of a negative number is negative.

Example:

\begin{matrix} \sqrt[3]{- 125} = \sqrt[3]{(- 5) \times (- 5) \times (- 5)} = - 5 \end{matrix}

$\begin{array}{l} \sqrt[3]{- 125} = \sqrt[3]{(- 5) \times (- 5) \times (- 5)} \\ = - 5 \end{array}$

4. To determine the cube roots of fractions, the fractions should be simplified to numerators and denominators that are cubes of integers.

Example:

\begin{matrix} \sqrt[3]{\frac{16}{250}} = \sqrt[3]{\frac{168}{250125}} = \sqrt[3]{\frac{8}{125}} = \frac{2}{5} \end{matrix}

$\begin{array}{l} \sqrt[3]{\frac{16}{250}} = \sqrt[3]{\frac{168}{250125}} \\ = \sqrt[3]{\frac{8}{125}} \\ = \frac{2}{5} \end{array}$

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