Operations with Radicals

Overview

Radicals, such as square roots, can be manipulated using various operations, including addition, subtraction, multiplication, and division. However, these operations follow specific rules to ensure proper simplification.

Key Rules for Operations with Radicals

  • Addition and Subtraction: Radicals can only be added or subtracted when they have the same index and radicand (the number inside the radical). For example:

    \(\sqrt{2} + 3\sqrt{2} = 4\sqrt{2}\)

  • Multiplication: Multiply the coefficients (numbers outside the radicals) and the radicands separately:

    \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\)

  • Division: Divide the coefficients and the radicands separately, and simplify if possible:

    \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\)

  • Always simplify radicals whenever possible by factoring out perfect squares.

Example 1: Adding Radicals

Simplify the following expression:

\(2\sqrt{3} + 4\sqrt{3}\)

Since the radicals have the same index and radicand, add the coefficients:

\((2 + 4)\sqrt{3} = 6\sqrt{3}\)

Example 2: Multiplying Radicals

Simplify the following expression:

\(\sqrt{5} \times \sqrt{7}\)

Multiply the radicands:

\(\sqrt{5 \times 7} = \sqrt{35}\)

Example 3: Dividing Radicals

Simplify the following expression:

\(\frac{\sqrt{50}}{\sqrt{2}}\)

Divide the radicands and simplify:

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

Practice Questions

  1. Question 1: Simplify the following expression:

    \(3\sqrt{7} + 5\sqrt{7}\)

    Solution

    Since the radicals are the same, add the coefficients:

    \((3 + 5)\sqrt{7} = 8\sqrt{7}\)

  2. Question 2: Simplify the following expression:

    \(\sqrt{12} \times \sqrt{3}\)

    Solution

    Multiply the radicands:

    \(\sqrt{12 \times 3} = \sqrt{36} = 6\)

  3. Question 3: Simplify the following expression:

    \(\frac{\sqrt{48}}{\sqrt{3}}\)

    Solution

    Divide the radicands and simplify:

    \(\sqrt{\frac{48}{3}} = \sqrt{16} = 4\)

  4. Question 4: Simplify the following expression:

    \(2\sqrt{5} - \sqrt{5}\)

    Solution

    Since the radicals are the same, subtract the coefficients:

    \((2 - 1)\sqrt{5} = \sqrt{5}\)

  5. Question 5: Simplify the following expression:

    \(\sqrt{8} \times \sqrt{2}\)

    Solution

    Multiply the radicands and simplify:

    \(\sqrt{8 \times 2} = \sqrt{16} = 4\)

→Introduction to Linear Relations