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.
Expand and simplify :
Problem 1 :
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