9.3 Add and Subtract Square Roots

Square roots are essential in algebra, letting us solve equations and simplify expressions. They're like a secret code that unlocks complex math problems. We'll learn how to add, subtract, and simplify square roots, making them easier to work with.

Understanding square roots helps us tackle irrational numbers and algebraic expressions. We'll explore how to combine similar radicals and use the distributive property with square roots. These skills are crucial for solving more advanced math problems down the road.

Operations with Square Roots

Addition of like square roots

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• Like square roots have the same radicand ($5$ in $\sqrt{5}$ and $\sqrt{5}$)
• Add or subtract coefficients (numbers before square root symbol) while keeping radicand unchanged
  • $2\sqrt{5} + 3\sqrt{5} = 5\sqrt{5}$
  • $7\sqrt{3} - 4\sqrt{3} = 3\sqrt{3}$
• Assume coefficient is 1 if not explicitly written ($\sqrt{7} + \sqrt{7} = 2\sqrt{7}$)

Simplification of radical expressions

• Factor out perfect squares from radicand ($\sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}$)
• Combine like terms in coefficient ($2\sqrt{3} + 5\sqrt{3} - \sqrt{3} = 6\sqrt{3}$)
• Multiply simplified radicand and coefficient for final expression ($2\sqrt{50} = 2\sqrt{25 \cdot 2} = 2 \cdot 5 \cdot \sqrt{2} = 10\sqrt{2}$)
• Simplification often involves reducing the radicand to its simplest form

Similar radicals in algebra

• Similar radicals have same index (small number in top left of radical symbol) and radicand
  • $2\sqrt{3}$ and $5\sqrt{3}$ are similar
• Add or subtract coefficients, keep radicand unchanged
  • $2\sqrt{x} + 5\sqrt{x} = 7\sqrt{x}$
  • $3\sqrt{2y} - \sqrt{2y} = 2\sqrt{2y}$
• Simplify resulting expression if possible
  • $2\sqrt{8x} + 3\sqrt{18x}$
    1. Factor out perfect squares: $2\sqrt{4 \cdot 2x} + 3\sqrt{9 \cdot 2x}$
    2. Simplify radicals: $2 \cdot 2\sqrt{2x} + 3 \cdot 3\sqrt{2x}$
    3. Multiply coefficients: $4\sqrt{2x} + 9\sqrt{2x}$
    4. Combine like terms: $13\sqrt{2x}$

Numbers and Square Roots

• Rational numbers are those that can be expressed as a ratio of two integers
• Irrational numbers cannot be expressed as a simple fraction and have non-repeating, non-terminating decimal representations
• Square roots of non-perfect squares are irrational numbers

Working with Algebraic Expressions

• Algebraic expressions often involve square roots
• The distributive property can be applied when multiplying a number or term by a square root expression
  • Example: $3(\sqrt{2} + \sqrt{5}) = 3\sqrt{2} + 3\sqrt{5}$