Multiplying And Dividing Radical Expressions - 2012 Book Archive

Multiplying Radical Expressions

When multiplying radical expressions with the same index, we use the product rule for radicals. If a and b represent positive real numbers,

 

Example 1: Multiply: 2⋅6.

Solution: This problem is a product of two square roots. Apply the product rule for radicals and then simplify.

Answer: 23

 

Example 2: Multiply: 93⋅63.

Solution: This problem is a product of cube roots. Apply the product rule for radicals and then simplify.

Answer: 3 23

 

Often there will be coefficients in front of the radicals.

 

Example 3: Multiply: 23⋅52.

Solution: Using the product rule for radicals and the fact that multiplication is commutative, we can multiply the coefficients and the radicands as follows.

Typically, the first step involving the application of the commutative property is not shown.

Answer: 106

 

Example 4: Multiply: −2 5x3⋅3 25x23.

Solution:

Answer: −30x

 

Use the distributive property when multiplying rational expressions with more than one term.

 

Example 5: Multiply: 43(23−36).

Solution: Apply the distributive property and multiply each term by 43.

Answer: 24−362

 

Example 6: Multiply: 4x23(2x3−5 4x23).

Solution: Apply the distributive property and then simplify the result.

Answer: 2x−10x⋅2x3

 

The process for multiplying radical expressions with multiple terms is the same process used when multiplying polynomials. Apply the distributive property, simplify each radical, and then combine like terms.

 

Example 7: Multiply: (5+2)(5−4).

Solution: Begin by applying the distributive property.

Answer: −3−25

 

Example 8: Multiply: (3x−y)2.

Solution:

Answer: 9x−6xy+y

 

Try this! Multiply: (23+52)(3−26).

Answer: 6−122+56−203

Video Solution

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The expressions (a+b) and (a−b) are called conjugatesThe factors (a+b) and (a−b) are conjugates.. When multiplying conjugates, the sum of the products of the inner and outer terms results in 0.

 

Example 9: Multiply: (2+5)(2−5).

Solution: Apply the distributive property and then combine like terms.

Answer: −3

 

It is important to note that when multiplying conjugate radical expressions, we obtain a rational expression. This is true in general and is often used in our study of algebra.

Therefore, for nonnegative real numbers a and b, we have the following property: