Square Roots and Cube Roots

Square Roots

The square root of a number is the value that, when squared, gives the original number. The square root is denoted as \sqrt{x}.

When we write the square root of a number, we are usually referring to both the positive and negative square roots. This is because a negative number squared is a positive number.

For example, since 3^2 = 9 and (-3)^2 = 9, we say \sqrt{9} = \pm 3.

Some examples of square roots are:

  • \sqrt{1} = \pm 1
  • \sqrt{4} = \pm 2
  • \sqrt{9} = \pm 3
  • \sqrt{16} = \pm 4

Approximating Square Roots

For non-perfect 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.

Let’s look at some examples.

Examples

Example 1:

\sqrt{13} \approx 3.605551275 (9 decimal places)

= 3.6 (1 decimal place)

= 3.61 (2 decimal places)

= 3.606 (3 decimal places)

Example 2:

\sqrt{13} \approx 3 (1 significant figure)

= 3.6 (2 significant figures)

= 3.61 (3 significant figures)

Cube Roots

The cube root of a number A is the value that, when cubed, gives the original number. The cube root is denoted as \sqrt[3]{A} or A^{\frac{1}{3}}.

Some examples of cube roots are:

\sqrt[3]{8} = 2

\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

Approximating Cube Roots

Just like square roots, cube roots of non-perfect cube numbers can be approximated to a specified number of decimal places or significant figures, as required by the problem.

Let’s look at some examples.

Examples

Example 1:

\sqrt[3]{50} \approx 3.684031498 (9 decimal places)

= 3.7 (1 decimal place)

= 3.68 (2 decimal places)

= 3.684 (3 decimal places)

Example 2:

\sqrt[3]{50} \approx 3.7 (1 significant figure)

= 3.68 (2 significant figures)

= 3.684 (3 significant figures)

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