2.1 Squares, Square Roots, Cube and Cube Roots

2.1 Squares, Square Roots, Cube and Cube Roots

(A) Squares
The square of a number is the answer you get when you multiply a number by itself.

Example:
(a) 13= 13 × 13 = 169
(b)   (–10)= (–10) × (–10) = 100
(c) (0.4)2 = 0.4 × 0.4 = 0.16
(d)   (–0.06)= (–0.06) × (–0.06) = 0.0036
$\begin{array}{l}\text{(e)}{\left(3\frac{1}{2}\right)}^{2}={\left(\frac{7}{2}\right)}^{2}=\frac{7}{2}×\frac{7}{2}=\frac{49}{4}\\ \left(\text{f}\right)\text{}{\left(-1\frac{2}{7}\right)}^{2}={\left(-\frac{9}{7}\right)}^{2}=\left(-\frac{9}{7}\right)×\left(-\frac{9}{7}\right)=\frac{81}{49}\end{array}$

(B) Perfect Squares
1. Perfect squares are the squares of whole numbers.

2. Perfect squares are formed by multiplying a whole number by itself.
Example:
4 = 2 × 2   9 = 3 × 3   16 = 4 × 4

3. The first twelve perfect squares are:
= 12, 22, 32, 42, 52, 62, 72, 82, 92, 102, 112, 122
= 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144

(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:
$\begin{array}{l}\text{(a)}\sqrt{169}=\sqrt{13×13}=13\\ \text{(b)}\sqrt{\frac{25}{64}}=\sqrt{\frac{5×5}{8×8}}=\frac{5}{8}\\ \text{(c)}\sqrt{\frac{72}{98}}=\sqrt{\frac{\overline{)72}36}{\overline{)98}49}}=\sqrt{\frac{6×6}{7×7}}=\frac{6}{7}\\ \text{(d)}\sqrt{3\frac{1}{16}}=\sqrt{\frac{49}{16}}=\frac{7}{4}=1\frac{3}{4}\\ \text{(e)}\sqrt{1.44}=\sqrt{1\frac{\overline{)44}11}{\overline{)100}25}}=\sqrt{\frac{36}{25}}=\frac{6}{5}=1\frac{1}{5}\end{array}$

(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. $"\sqrt[3]{}"$  is the symbol for cube root.
Example:
$\begin{array}{l}\sqrt[3]{64}=\sqrt[3]{4×4×4}\\ \text{}=4\end{array}$
$\sqrt[3]{64}$ is read as ‘cube root of sixty-four’.

2.
The cube root of a positive number is positive.
Example:
$\begin{array}{l}\sqrt[3]{125}=\sqrt[3]{5×5×5}\\ \text{}=5\end{array}$

3.
The cube root of a negative number is negative.
Example:
$\begin{array}{l}\sqrt[3]{-125}=\sqrt[3]{\left(-5\right)×\left(-5\right)×\left(-5\right)}\\ \text{}=-5\end{array}$

4.
To determine the cube roots of fractions, the fractions should be simplified to numerators and denominators that are cubes of integers.
Example:
$\begin{array}{l}\sqrt[3]{\frac{16}{250}}=\sqrt[3]{\frac{\overline{)16}8}{\overline{)250}125}}\\ \text{}=\sqrt[3]{\frac{8}{125}}\\ \text{}=\frac{2}{5}\end{array}$