Simplifying Radical Expressions

Simplifying Radical Expressions

Combining Like Radicals

When adding or subtracting radical expressions, it is important to simplify each radical and combine like terms where possible. Like radicals have the same radicand (the number under the radical sign).

• Radical Expression: An expression that contains a square root, cube root, or higher root symbol.

• Like Radicals: Radicals with the same index and the same radicand.

Example Problem

Simplify the following expression:

$6\sqrt{125} - 2\sqrt{45}$

• Step 1: Simplify each radical.

  • $\sqrt{125} = \sqrt{25 \times 5} = \sqrt{25} \times \sqrt{5} = 5\sqrt{5}$

  • $\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}$

• Step 2: Substitute the simplified radicals back into the expression.

  • $6\sqrt{125} - 2\sqrt{45} = 6(5\sqrt{5}) - 2(3\sqrt{5})$

  • $= 30\sqrt{5} - 6\sqrt{5}$

• Step 3: Combine like terms.

  • $30\sqrt{5} - 6\sqrt{5} = (30 - 6)\sqrt{5} = 24\sqrt{5}$

Final Answer:

$24\sqrt{5}$

Key Points

• Always simplify radicals before combining.

• Only like radicals (same radicand and index) can be combined by adding or subtracting their coefficients.

Additional info:

• This process is foundational for algebraic manipulation in precalculus and higher mathematics.