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Detailed Explanation of Adding and Subtracting Surds

Understanding the Concept

Surds are irrational numbers expressed in their root form, such as 2\sqrt{2} or 5\sqrt{5}, which cannot be simplified into exact decimals. When adding or subtracting surds, specific rules must be followed to ensure correct simplification.

1. The Importance of Like Terms

Just as in algebra, you can only add or subtract surds if they are like terms—meaning they have the same number under the square root sign.

• Example 1: 25+75=952\sqrt{5} + 7\sqrt{5} = 9\sqrt{5}, since both terms contain 5\sqrt{5}.
• Example 2: 25+762\sqrt{5} + 7\sqrt{6} cannot be simplified further, as 5\sqrt{5} and 6\sqrt{6} are different surds.

2. The Need for Simplification

In many cases, surds must be simplified before they can be added or subtracted. This involves breaking them down into factors, identifying perfect square factors, and simplifying accordingly.

How to Simplify Surds Before Addition or Subtraction

1. Identify perfect square factors: Find square numbers within the radicand (the number inside the square root).
2. Rewrite the surd: Express the surd in terms of the square root of its factors.
3. Extract the square root of perfect squares: Factor out and simplify.
4. Combine like terms: Add or subtract the surds if they are now similar.
5. Leave unlike terms separate: If surds remain different, they cannot be combined.

Examples of Adding and Subtracting Surds

Example 1: Simplify 102−7210\sqrt{2} – 7\sqrt{2}

Since both terms contain 2\sqrt{2}, simply subtract the coefficients:

102−72=3210\sqrt{2} – 7\sqrt{2} = 3\sqrt{2}

Example 2: Simplify 6246\sqrt{24}

24=4×6=4×6=26\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}

Now multiply by 6:

6×26=1266 \times 2\sqrt{6} = 12\sqrt{6}

Example 3: Simplify 27−12\sqrt{27} – \sqrt{12}

27=9×3=33\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3} 12=4×3=23\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}

Subtracting:

33−23=33\sqrt{3} – 2\sqrt{3} = \sqrt{3}

Example 4: Simplify 63+28−175\sqrt{63} + \sqrt{28} – \sqrt{175}

63=9×7=37\sqrt{63} = \sqrt{9 \times 7} = 3\sqrt{7} 28=4×7=27\sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7} 175=25×7=57\sqrt{175} = \sqrt{25 \times 7} = 5\sqrt{7}

Now, add and subtract:

37+27−57=03\sqrt{7} + 2\sqrt{7} – 5\sqrt{7} = 0

Example 5: Simplify 20+45+48\sqrt{20} + \sqrt{45} + \sqrt{48}

20=4×5=25\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5} 45=9×5=35\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5} 48=16×3=43\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}

Combining like terms:

25+35+43=55+432\sqrt{5} + 3\sqrt{5} + 4\sqrt{3} = 5\sqrt{5} + 4\sqrt{3}

Since 5\sqrt{5} and 3\sqrt{3} are not like terms, they remain separate.

3. Rules to Remember When Working with Surds

• Only like surds can be added or subtracted.
• Always simplify surds first if possible.
• Break down the radicand into perfect squares to simplify further.
• Leave unlike surds as separate terms.
• Follow basic surd rules, such as ab=a2ba\sqrt{b} = \sqrt{a^2b}, which helps in breaking down complex expressions.

Summary of Steps for Adding and Subtracting Surds

1. Check if the surds have the same radicand.
2. Simplify each surd individually by breaking them into their prime factors.
3. Extract perfect square factors and simplify.
4. Combine like surds by adding or subtracting coefficients.
5. Leave the final answer in its simplest form.

Mastering these techniques allows for efficient manipulation of surds in mathematical problems, including algebra, geometry, and higher-level mathematics.