Radicals: Introduction to Square Roots

Overview

A square root is a number that, when multiplied by itself, equals the original number. The square root of a number a is written as:

\(\sqrt{a}\)

For example:

  • \(\sqrt{9} = 3\), because \(3 \times 3 = 9\).
  • \(\sqrt{16} = 4\), because \(4 \times 4 = 16\).

Key Points

  • Every positive number has two square roots: a positive root and a negative root. For example, the square roots of 9 are \(3\) and \(-3\), because both satisfy \(x^2 = 9\).
  • The principal square root (the positive root) is typically used unless stated otherwise.
  • Square roots are only defined for non-negative numbers in the set of real numbers.

Steps to Simplify Square Roots

  1. Identify if the number is a perfect square (e.g., 4, 9, 16, 25, etc.).
  2. If it is a perfect square, find the number that multiplies by itself to give the original number.
  3. If it is not a perfect square, estimate or leave the result as a radical (e.g., \(\sqrt{7}\)).

Example 1: Perfect Square

Find the square root of 25.

\(\sqrt{25} = 5\)

Explanation: \(5 \times 5 = 25\), so \(\sqrt{25} = 5\).

Example 2: Non-Perfect Square

Simplify \(\sqrt{20}\).

\(\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}\)

Explanation: Break \(20\) into factors where one is a perfect square, then simplify.

Practice Questions

  1. Question 1: Find the square root of 36.
    Solution

    \(\sqrt{36} = 6\), because \(6 \times 6 = 36\).

  2. Question 2: Simplify \(\sqrt{50}\).
    Solution

    \(\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}\)

  3. Question 3: Simplify \(\sqrt{72}\).
    Solution

    \(\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}\)

  4. Question 4: Find the square root of 121.
    Solution

    \(\sqrt{121} = 11\), because \(11 \times 11 = 121\).

  5. Question 5: Simplify \(\sqrt{32}\).
    Solution

    \(\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}\)

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