2.1 Squares, Square Roots, Cube and Cube Roots

Posted on June 10, 2017 by user

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) 13²= 13 × 13 = 169

(b) (–10)²= (–10) × (–10) = 100

(d) (–0.06)²= (–0.06) × (–0.06) = 0.0036

\begin{array}{l} (e) {(3 \frac{1}{2})}^{2} = {(\frac{7}{2})}^{2} = \frac{7}{2} \times \frac{7}{2} = \frac{49}{4} \\ (f) {(- 1 \frac{2}{7})}^{2} = {(- \frac{9}{7})}^{2} = (- \frac{9}{7}) \times (- \frac{9}{7}) = \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} (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]{} "

is the symbol for cube root.

Example:

\begin{array}{l} \sqrt[3]{64} = \sqrt[3]{4 \times 4 \times 4} \\ = 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 \times 5 \times 5} \\ = 5 \end{array}

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

Example:

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