Multiplying Radicals
Multiplying Radicals Multiplying One-term Radical Expressions Generalized Product Rule of Radicals √
√
√
Example 1 √
√
√
Apply product rule Multiply outside and inside the radical
√
Simplify the radical
√
Take the square root where possible
√
Simplify coefficients
√
Simplified expression
√
Example 2 √
√
√
√
Apply product rule Multiply outside and inside the radical Simplify the radical, index of 3
√ √ √ √
Multiply coefficients Simplified expression
Multiplying Radicals Multiplying Multi-term Radical Expressions Multiplying Multi-term Radical Expressions Each term of the first expression is multiplied by each term of the second polynomial.
Example 3 √
√ ( √
√
√ ( √ √ √
)
Distribute √ to each term of the second expression, apply product rule
)
√
Simplify each radical
√
Take square root where possible
√
Multiply coefficients
√
√
No like terms Simplified expression
√
Example 4 (√
(√
√ )( √
√ )( √
√ )
√ )
FOIL, following rules for multiplying radicals
√
√
√
Simplify radicals, find perfect square factors
√
√
√
Take square root where possible
√
√
√
√
√
√
√ √
√
√
√ √
Multiply coefficients Combine like terms Our Solution
Multiplying Radicals Example 5 ( √
( √
√ )( √ √
√
√ )( √
√
√
√
√
FOIL, following rules for multiplying radicals
√ ) √
Simplify radicals, find perfect square factors
√
√ √
√
√
√ )
√
√
Take square root where possible
√
Multiply coefficient No like terms Simplified expression
√
Example 6 – Multiplying Conjugates*
(
√ √ √
(
√
)(
)(
√
)
√ √
√
√
)
FOIL, Apply product rule for radicals Simplify radicals Combine like terms
Simplified expression *This is an important case. Notice that when we multiply these conjugates together, we do not have a radical in our final product. This will be important when rationalizing denominators.
√
√
Example 1 √
√
√
Apply product rule Multiply outside and inside the radical
√
Simplify the radical
√
Take the square root where possible
√
Simplify coefficients
√
Simplified expression
√
Example 2 √
√
√
√
Apply product rule Multiply outside and inside the radical Simplify the radical, index of 3
√ √ √ √
Multiply coefficients Simplified expression
Multiplying Radicals Multiplying Multi-term Radical Expressions Multiplying Multi-term Radical Expressions Each term of the first expression is multiplied by each term of the second polynomial.
Example 3 √
√ ( √
√
√ ( √ √ √
)
Distribute √ to each term of the second expression, apply product rule
)
√
Simplify each radical
√
Take square root where possible
√
Multiply coefficients
√
√
No like terms Simplified expression
√
Example 4 (√
(√
√ )( √
√ )( √
√ )
√ )
FOIL, following rules for multiplying radicals
√
√
√
Simplify radicals, find perfect square factors
√
√
√
Take square root where possible
√
√
√
√
√
√
√ √
√
√
√ √
Multiply coefficients Combine like terms Our Solution
Multiplying Radicals Example 5 ( √
( √
√ )( √ √
√
√ )( √
√
√
√
√
FOIL, following rules for multiplying radicals
√ ) √
Simplify radicals, find perfect square factors
√
√ √
√
√
√ )
√
√
Take square root where possible
√
Multiply coefficient No like terms Simplified expression
√
Example 6 – Multiplying Conjugates*
(
√ √ √
(
√
)(
)(
√
)
√ √
√
√
)
FOIL, Apply product rule for radicals Simplify radicals Combine like terms
Simplified expression *This is an important case. Notice that when we multiply these conjugates together, we do not have a radical in our final product. This will be important when rationalizing denominators.
