Square and square root

3.1 Squares and Square Roots

3.1

Squares and Square Roots

Squares:

	Definition

	A number that indicates a number is multiplied by the number itself.

Examples: \(2^2,\, 9^2,\, 5^2\)

Perfect squares:

	Defintion

	A number that is equal to the square of a whole number.

Examples: \(1,\, 4,\, 9\)

Determine a number is a perfect square:

Perfect square can be written as a product of two equal factors.

Example

\(\begin{aligned} 144&=12\times12 \\\\&=12^2. \end{aligned}\)

\(144\) is a perfect square.

Relationship between squares and square roots:

Finding the square and finding the square root are inverse operations.

Example

The square of \(8\) is \(64\).

The square root of \(64\) is \(8\).

\(8\times8=64\)

Thus,

\(\begin{aligned} \sqrt{64}&=\sqrt{8\times8} \\\\&=8. \end{aligned}\)

The square of a number:

Example

Calculate:

(i)

\(\begin{aligned} 3^2&=3\times3 \\\\&=9. \end{aligned}\)

(ii)

\(\begin{aligned} \bigg(\dfrac{2}{5}\bigg)^2&=\dfrac{2}{5}\times\dfrac{2}{5} \\\\&=\dfrac{4}{25}. \end{aligned}\)

The square root of a number:

Example

Solve:

(i)

\(\begin{aligned} \sqrt{121}&=\sqrt{11\times11} \\\\&=\sqrt{11^2}\\\\&=11. \end{aligned}\)

(ii)

\(\begin{aligned} \sqrt{\dfrac{25}{49}}&=\sqrt{\dfrac{5}{7}\times\dfrac{5}{7}} \\\\&=\sqrt{\bigg(\dfrac{5}{7}\bigg)^2} \\\\&=\dfrac{5}{7}. \end{aligned}\)

Generalisation when two square roots are multiplied:

Square roots of the same numbers

The product of two equal square root numbers is the number itself.
\(\sqrt{a}\times\sqrt{a}=a\)

Square roots of different numbers

The product of two different square root numbers is the square root of the product of the two numbers.
\(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\)

Quick Notes