Square Roots and Cube Roots

Square Roots

The square root of a non-negative real number A is written as \sqrt{A}, such that

\sqrt{A}\times \sqrt{A}=\left( \sqrt{A}\right) ^{2}

= A

Since 3\times 3=9, the square root of 9 is \sqrt{9}=3.

When we write the square root of a number, we are usually providing both the positive square root amd the negative square root. This is because a negative number squared is a positive number.

3\times 3=9.

So, \sqrt{9}=\pm 3

Examples

1. \sqrt{13}= 3.605551275 (9 decimal places)

= 3.6 (1 decimal place)

= 3.61 (2 decimal places)

= 3.606 (3 decimal places)

2. \sqrt{13}= 3 (1 significant figure)

= 3.6 (2 significant figures)

= 3.61 (3 significant figures)

Cube Roots

The cube root of A is written as \sqrt[3] {A} or A^{\dfrac{1}{3}}.

Since the cube of 2 is 2^{3}=8, we say \sqrt[3] {8}=2

We can use a calculator to find the square root or cube root of a number. For non-perfect square numbers, the square numbers, the square roots can be given correct to a certain number of decimal places, or correct to a certain number of significant figures. The problem will state the accuracy to be given for the answer.

Examples

\sqrt [ 3 ]{ 343 } =7

\sqrt [ 3 ]{ 8000 } =20

\sqrt [ 3 ]{ 27000 } =30

\sqrt [ 3 ]{ 1 } =1

\sqrt [ 3 ]{ -8 } =-2

\sqrt [ 3 ]{ 0.008 } =0.2