Module 3: Multiplying Radical Expressions

Haberman MTH 95

Section IV: Radical Expressions, Equations, and Functions

Module 3: Multiplying Radical Expressions

Recall the property of exponents that states that ambm = (ab)m . We can use this rule to

obtain an analogous rule for radicals:

n a n b = a1 n b1 n = (ab)1 n (using the property of exponents given above) = n ab

Product Rule for Radicals

If a and b are positive real numbers and n is a positive integer, then n a n b = n ab .

EXAMPLE: Perform the indicated multiplication, and simplify completely.

a. 2 18

b. 4 3x2 4 27x2

SOLUTIONS:

a. 2 18 = 2 18 (product rule for radicals)

= 36

= 62 =6

(write 36 as a perfect square)

b. 4 3x2 4 27x2 = 4 3x2 27x2 (product rule for radicals) = 4 3 27 x2 x2 = 4 81x4 = 4 81 4 x4 (product rule for radicals) = 3 x (we need to use the absolute value since 4 is even)

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Product Rule for Simplifying Radical Expressions: When simplifying a radical expression it is often necessary to use the following equation which is equivalent to the product rule:

n ab = n a n b .

EXAMPLE: Simplify 40 .

SOLUTION: Since 40 isn't a perfect square, we need to write 40 as a product containing a

factor that is a perfect square:

40 = 4 10 = 4 10 = 2 10

(factor 40 using perfect square(s)) (product rule for simplifying radical expressions)

EXAMPLE: Simplify the following.

a. 3 24

b. 4 16w8

c. 54d 5

SOLUTIONS:

a. 3 24 = 3 83 (factor 24 using perfect cube(s)) = 3 8 3 3 (product rule for simplifying radical expressions) = 233

b. 4 16w8 = 4 16 4 w8 (product rule for simplifying radical expressions)

( ) = 4 24 4 w2 4

= 2w2 (we don't need the absolute value here since w2 must be positive)

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c. 54d 5 = 9 6 d 4 d = 9 6 d4 d = 3d 2 6d

(product rule for simplifying radical expressions)

Try these yourself and check your answers. Perform the indicated multiplication, and simplify completely.

a. 14 21 .

b. 3 3y2 3 9 y .

SOLUTIONS:

a. 14 21 = 14 21 = 2737 = 72 6 =7 6

b. 3 3y2 3 9 y = 3 3y2 9 y = 3 27 y3 = 3 33 y3 = 3y

EXAMPLE: Perform the following multiplication: 3 x 4 x .

SOLUTION:

The key step when the indices of the radicals are different is to write the expressions with rational exponents.

3 x 4 x = x1 3 x1 4 (write with rational exponents)

= x1 3 + 1 4 (use a propery of exponents)

= x4 12 + 3 12

(create a common denominator for the exponent)

= x7 12

(use another property of exponents)

= 12 x7 (write final answer in radical form to agree with original expression)

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Try these yourself and check your answers. Perform the indicated multiplication, and simplify completely.

a. t 8 t3

b. 3 2 p2 3 p

SOLUTIONS:

a. t 8 t3 = t1 2 t3 8 = t1 2 + 3 8 = t4 8 + 3 8

(write with rational exponents) (use a property of exponents) (create a common denominator for the exponents)

= t78

(use another property of exponents)

= 8 t7

(write final answer in radical notation to agree with the original expression)

( ) b. 3 2 p2 3 p = 2 p2 1 3 (3 p)1 2 (write with rational exponents) ( ) = 2 p2 2 6 (3 p)3 6 (create a common denominator for the exponents)

( ) ( ) = 2 p2 2 1 6 (3 p)3 1 6

( ) = 4 p4 27 p3 1 6 ( ) = 108 p7 1 6

= 6 108 p7

= 6 108 p6 p

= p 6 108 p

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