**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:

Example:

^{2 }= 13 × 13 =

**169**(a) 13

^{2 }= (–10) × (–10) =

**100**(b) (–10)

^{2}= 0.4 × 0.4 =

**0.16**(c) (0.4)

^{2 }= (–0.06) × (–0.06) =

**0.0036**(d) (–0.06)

**(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:

= 1

^{2}, 2^{2}, 3^{2}, 4^{2}, 5^{2}, 6^{2}, 7^{2}, 8^{2}, 9^{2}, 10^{2}, 11^{2}, 12^{2}**= 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:**

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

**(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

3

^{3}= 3 × 3 × 3 = 27

**The cube of a negative number is negative.**

2.

2.

*Example:*(–2)

^{3}= (–2) × (–2) × (–2)= –8

**3.**The cube of zero is zero. The cube of one is one, 1

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

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

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

2.

2.

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

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

3.

3.

*Example:*
$\begin{array}{l}\sqrt[3]{-125}=\sqrt[3]{\left(-5\right)\times \left(-5\right)\times \left(-5\right)}\\ \text{}=-5\end{array}$

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

4.

4.

*Example:*
$\begin{array}{l}\sqrt[3]{\frac{16}{250}}=\sqrt[3]{\frac{\overline{)16}8}{\overline{)250}125}}\\ \text{}=\sqrt[3]{\frac{8}{125}}\\ \text{}=\frac{2}{5}\end{array}$