What is how to multiply radicals?

Multiplying Radicals

Multiplying radicals involves combining terms under the radical sign and simplifying the result. Here's a breakdown of the process:

1. Basic Principle:

The fundamental rule for multiplying radicals is:

√a * √b = √(a * b)

This rule holds true when 'a' and 'b' are non-negative numbers.

2. Multiplying Radicals with the Same Index:

• Multiply the coefficients: If there are coefficients (numbers in front of the radical), multiply them together.
• Multiply the radicands: Multiply the numbers or expressions inside the radical symbols.
• Simplify the Result: After multiplying, simplify the resulting radical. Look for perfect squares, cubes, or higher powers (depending on the index) within the radicand and extract them. See Simplifying Radicals

Example:

3√2 * 5√3 = (3 * 5)√(2 * 3) = 15√6

3. Multiplying Radicals with Different Indices:

To multiply radicals with different indices (e.g., square root and cube root), you must first convert them to radicals with a common index.

• Find the Least Common Multiple (LCM) of the indices: This will be the new index for both radicals.
• Adjust the Radicands: Raise each radicand to the power that will make the index match the LCM.
• Multiply and Simplify: Once the indices are the same, you can multiply the radicals as described above and simplify the resulting radical.

Example:

√2 * ∛3 (Square root has an index of 2, cube root has an index of 3)

• LCM of 2 and 3 is 6.
• √2 = 2^1/2 = 2^3/6 = ⁶√2³ = ⁶√8
• ∛3 = 3^1/3 = 3^2/6 = ⁶√3² = ⁶√9
• ⁶√8 * ⁶√9 = ⁶√(8 * 9) = ⁶√72

4. Multiplying Radicals with Variables:

The same principles apply when multiplying radicals containing variables. Remember to apply the rules of exponents when multiplying variables under the radical. See Radicals and Exponents

Example:

√(2x) * √(8x³) = √(2x * 8x³) = √(16x⁴) = 4x²

5. Distributive Property:

When multiplying a radical expression by a sum or difference of terms, use the distributive property.

Example:

√2 * (√3 + √5) = (√2 * √3) + (√2 * √5) = √6 + √10

6. FOIL Method:

When multiplying two binomials containing radicals, use the FOIL (First, Outer, Inner, Last) method. See FOIL Method

Example:

(√2 + 1)(√3 - 2) = (√2 * √3) + (√2 * -2) + (1 * √3) + (1 * -2) = √6 - 2√2 + √3 - 2

Important Considerations:

• Always simplify radicals before and after multiplying to make the process easier.
• Pay attention to the index of the radical.
• Remember the rules of exponents when multiplying variables.