Introduction to Surds, Simplification, and Comparison

Bymypaperr

1. Lesson Overview

In the Grade 11 syllabus, a major portion of the Real Numbers unit focuses on surds. Learning how to simplify surds and convert between entire surds and mixed surds is essential before performing algebraic operations.

2. Definitions and Key Forms

A. Surd

Definition: A surd is an irrational root of a positive rational number that cannot be simplified into an exact whole number or fraction.

Examples of Surds: \(\sqrt{2}, \; \sqrt{3}, \; \sqrt{5}, \; \sqrt{10}\)

Not Surds: \(\sqrt{4} \; (= 2), \; \sqrt{9} \; (= 3), \; \sqrt{\frac{1}{25}} \; (= \frac{1}{5})\) because their exact values are rational.

B. Order of a Surd

In the expression \(\sqrt[n]{a}\), \(n\) is called the order of the surd and \(a\) is the radicand.

In the G.C.E. O/L syllabus, we study quadratic surds (surds of order \(2\), written simply as \(\sqrt{a}\)).

C. Entire Surds vs. Mixed Surds

Entire Surd: A surd where all factors are placed inside the square root sign (the coefficient outside is \(1\)).
Examples: \(\sqrt{8}, \; \sqrt{18}, \; \sqrt{50}\)

Mixed Surd: A surd written as the product of a rational number (other than \(1\)) and a surd.
Examples: \(2\sqrt{2}, \; 3\sqrt{2}, \; 5\sqrt{2}\)

D. Fundamental Multiplication Rule for Surds

For any two positive real numbers \(a\) and \(b\):

\[\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\]

3. Worked Examples (Step-by-Step)

Example 1: Simplifying Entire Surds into Mixed Surds

Express \(\sqrt{72}\) in its simplest mixed surd form.

Solution:

Step 1: Find the largest perfect square factor of \(72\). (Perfect squares are \(4, 9, 16, 25, 36, 49, \dots\))
\[72 = 36 \times 2\]
Step 2: Rewrite the square root as a product of two roots.
\[\sqrt{72} = \sqrt{36 \times 2}\] \[\sqrt{72} = \sqrt{36} \times \sqrt{2}\]
Step 3: Take the square root of the perfect square (\(36\)).
\[\sqrt{72} = 6 \times \sqrt{2}\] \[\sqrt{72} = 6\sqrt{2}\]

Example 2: Simplifying Entire Surds

Simplify \(\sqrt{125}\).

Solution:

Step 1: Factorize \(125\) using its largest perfect square factor (\(25\)).
\[125 = 25 \times 5\]
Step 2: Apply the multiplication rule \(\sqrt{ab} = \sqrt{a} \times \sqrt{b}\).
\[\sqrt{125} = \sqrt{25 \times 5}\] \[\sqrt{125} = \sqrt{25} \times \sqrt{5}\]
Step 3: Simplify \(\sqrt{25}\).
\[\sqrt{125} = 5\sqrt{5}\]

Example 3: Converting Mixed Surds into Entire Surds

Express \(4\sqrt{3}\) as an entire surd.

Solution:

Step 1: Square the rational coefficient outside the root (\(4\)) and place it under a square root sign.
\[4 = \sqrt{4^2} = \sqrt{16}\]
Step 2: Multiply the two roots together.
\[4\sqrt{3} = \sqrt{16} \times \sqrt{3}\] \[4\sqrt{3} = \sqrt{16 \times 3}\] \[4\sqrt{3} = \sqrt{48}\]

Example 4: Comparing the Magnitude of Surds (O/L Exam Standard)

Arrange the following surds in ascending order (smallest to largest):

\[3\sqrt{2}, \quad 2\sqrt{5}, \quad \sqrt{19}\]

Solution:

Step 1: Convert all numbers into entire surds so they can be compared easily.

For \(3\sqrt{2}\):

\[3\sqrt{2} = \sqrt{3^2 \times 2}\] \[3\sqrt{2} = \sqrt{9 \times 2}\] \[3\sqrt{2} = \sqrt{18}\]

For \(2\sqrt{5}\):

\[2\sqrt{5} = \sqrt{2^2 \times 5}\] \[2\sqrt{5} = \sqrt{4 \times 5}\] \[2\sqrt{5} = \sqrt{20}\]

For \(\sqrt{19}\): \(\sqrt{19}\) is already an entire surd.

Step 2: Compare the numbers inside the square root signs.
\[18 < 19 < 20\] \[\sqrt{18} < \sqrt{19} < \sqrt{20}\]
Step 3: Write the final answer using the original surds given in the question.
\[3\sqrt{2}, \quad \sqrt{19}, \quad 2\sqrt{5}\]

4. Practice Exercises for Students

  1. Express each of the following as a mixed surd in its simplest form:
    1. \(\sqrt{45}\)
    2. \(\sqrt{98}\)
    3. \(\sqrt{200}\)
  2. Express each of the following as an entire surd:
    1. \(5\sqrt{2}\)
    2. \(3\sqrt{7}\)
    3. \(6\sqrt{3}\)
  3. Arrange \(4\sqrt{3}\), \(5\sqrt{2}\), and \(\sqrt{47}\) in descending order (largest to smallest).
Show Answers

1. (a) \(3\sqrt{5}\)    (b) \(7\sqrt{2}\)    (c) \(10\sqrt{2}\)

2. (a) \(\sqrt{50}\)    (b) \(\sqrt{63}\)    (c) \(\sqrt{108}\)

3. \(5\sqrt{2}, \quad 4\sqrt{3}, \quad \sqrt{47}\)   (since \(\sqrt{50} > \sqrt{48} > \sqrt{47}\))

Similar Posts

Leave a Reply

Your email address will not be published. Required fields are marked *