USING DISTRIBUTIVE PROPERTY WITH RATIONAL EXPONENTS

For example, 

a(b + c) = ab + ac

When it involves rational exponents, we have to know about rules in exponents.

productruleofexponents

Expand and simplify :

Problem 1 :

2x123x32 - 4x-12

Solution :

2x123x32 - 4x-12

Use the distributive property.

=2x12·3x32 - 2x12·4x-12

Using properties of exponents, we get

= 6x12 + 32 - 8x12 + -12= 6x42 - 8x12 - 12= 6x2 - 8x0= 6x2 - 8

Problem 2 :

x12x12 - x-12

Solution :

x12x12 - x-12

Use the distributive property.

=x12·x12 - x12·x-12

Using properties of exponents, we get

= x12 + 12 - x12 + -12= x22 - x12 - 12= x1 - x0= x - 1

Problem 3 :

x-12x12 - 2x-12

Solution :

x-12x12 - 2x-12

Use the distributive property.

=x-12·x12 - x-12·2x-12

Using properties of exponents, we get

= x-12 + 12 - 2x-12 + -12= x0 - 2x-12 - 12= 1 - 2x-22= 1 - 2x-1= 1 - 2 ·1x= 1 - 2x= x - 2x

Problem 4 :

x322x12 - x-12

Solution :

x322x12 - x-12

Use the distributive property.

=x32·2x12 - x32·x-12

Using properties of exponents, we get

= 2x32 + 12 - x32 + -12= 2x42 - x32 - 12= 2x2 - x22= 2x2 - x1= 2x2 - x= x(2x - 1)

Problem 5 :

x12 + 3 x12 - 3

Solution :

x12 + 3 x12 - 3

Use the distributive property.

= x12 · x12 - 3 · x12 + 3 · x12 - 3 · 3

Using properties of exponents, we get

= x12 + 12 - 9= x22 - 9= x - 9

Problem 6 :

x12 + x-12 x12 - x-12

Solution :

x12 + x-12 x12 - x-12

Use the distributive property.

= x12 · x12 - x12 · x-12 + x-12 · x12 - x-12 · x-12

Using properties of exponents, we get

= x12 · x12 - x-12 · x-12= x12 + 12 - x-12 - 12= x22 - x-22= x1 - x-1= x - 1x= x2 - 1x

Problem 7 :

x + 3x2

Solution :

x + 3x2

Use the distributive property.

= x + 3x·x + 3x

Using properties of exponents, we get

= x2 + x 3x + x 3x + 3x · 3x= x2 + 3 + 3 + 9x2= x2 + 9x2 + 6

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