Question

Applying properties of Exponents in exercise,use the properties of exponents to simplify the expression.See example 1.

(a) 5^3/25^2
(b) (92/3)(3)(32/3)
(c) [(251/2)(52)]1/3
(d) (82)(43)

Answer 1

Answer: a)
`(1)/(5)` b) 27 c) 5 d)
`2^(12)=4096`

Explanation:

Properties of exponent :

1. 
   `a^m* a^n= a^(m+n)`
2. 
   `(a^m)/(a^n)=a^(m-n)`
3. 
   `(a^m)^n=a^(mn)`
4. 
   `a^(-n)=(1)/(a^n)`

By using above properties we simplify the given expressions.

a)
`(5^3)/(25^2)=(5^3)/((5^2)^2)`

`=(5^3)/(5^4)` [ By using (3)]

`=5^(3-4)=5^(-1)` [ By using (2)]

`=(1)/(5)` [ By using (4)]

b)
`(9^{(2)/(3)})(3)(3^{(3)/(2)})`

`=((3^(2))^{(2)/(3)})(3^{1+(3)/(2)})` [ By using (1)]

`=(3^{(4)/(3)})(3^{(5)/(2)})` [ By using (3)]

`=3^{(4)/(3)+(5)/(3)}` [ By using (1)]

`=3^{(4+5)/(3)}=3^{(9)/(3)=3^3=27}`

c)
`[(25)^(1)/(2)(5^2)]^{(1)/(3)} =[(5^2)^(1)/(2)(5^2)]^(1)/(3)`

`=[(5)(5^2)]^(1)/(3)` [ By using (3)]

`=[5^(1+2)]^(1)/(3)` [ By using (1)]

`=5^{3*(1)/(3)})= 5` [ By using (3)]

d)
`(8^2)(4^3)= (2^3)^2((2^2)^3)` [ By using (5)]

`=(2^6)(2^6)` [ By using (3)]

`=2^(6+6)=2^(12)=4096`