Exponents and Power

Exercise 12.1

Question 1: Evaluate.

(i) 3^–2

Answer: 3^-2 `=(1)/(3^2)=1/9`

(ii) `(– 4)^(– 2)`

Answer: `(-4)^(-2)=(1)/(-4)^2=(1)/(16)`

(iii) `(1/2)^(-2)+(1/3)^(-2)+(1/4)^(-2)`

Answer: = 2² + 3² + 4² = 4 + 9 + 16 = 29

Question 2: Simplify and express the result in power notation with positive exponent.

(i) `(-4)^5÷(-4)^8`

Answer: `a^m÷a^n=a^(m-n)`

So, `(-4)^5÷(-4)^8`

`=(-4)^(5-3)=(-4)^(-3)`

(ii) `((1)/(2^3))^2`

Answer: `(a^m)^n=a^(mn)`

So, `((1)/(2^3))^2`

`=1/(2^6)=1/(64)`

(iii) `-3^4xx(5/3)^4`

Answer: Since power on 3 is even so `-3^4=3^4`

So, given expression can be written as follows:

`3^4xx3^(-4)xx5^4`

`=3^(4-4)xx5^4`

`=3^0xx5^4=1xx5^4=625`

(iv) `(3^(-7)÷3^(-10))xx3^(-5)`

Answer: `3^(-7+10-5)=3^(-2)=((1)/(3^2))`

(v) `2^(-3)xx (-7)^(-3)`

Answer: `((1)/(2^3))xx((1)/(-7)^3)`

`=((1)/(-14)^3)`

Question 3: Find the value of.

(i) `(3^0 + 4^(-1)) xx 2^2`

Answer: `(3^0 + 4^(-1)) xx 2^2`

`=(1+1/4)xx4=5/4xx4=5`

(ii) `(2^(-1) xx 4^(-1)) ÷ 2^(-2)`

Answer: `=(1/2xx1/4)÷1/4`

`=1/2xx1/4xx4=1/2`

(iii) `(1/2)^(-2)+(1/3)^(-2)+(1/4)^(-2)`

Answer: `=2^2+3^2+4^2`

`=4+9+16=29`

(iv) `(3^(-1)+4^(-1)+(5^-1)^0`

Answer: We know that `a^0=1`

So, `(3^(-1)+4^(-1)+(5^-1)^0=1`

(v) `[((-2)/(3))^(-2)]^2`

Answer: `=((3^2)/(2^2))^2`

`=(3^4)/(2^4)=(81)/(16)`

Question 4: Evaluate

(i) `(8^(-1)xx5^3)/(2^(-4))`

Answer: `=((2^3)^(-1)xx5^3)/(2^(-4))`

`=2^(-3)xx2^4xx5^3`

`=2^(-3+4)xx5^3=2^1xx5^3=5^3`

(ii) `(5^(-1)xx2^(-1))xx6^(-1)`

Answer: `=1/5xx1/2xx1/6=(1)/(60)`

Question 5: Find the value of m for which `5^m ÷ 5^(-3) = 5^5`

Answer: `a^5 = a^(m-n)`
Here, `m-n = 5` and `n = -3`
So, `m = 5+(-3)=2`

Question 6: Evaluate

(i) `((1/3)^(-1)-(1/4)^(-1))^(-1)`

Answer: `=(3-4)^(-1)=-1^(-1)=-1`

(ii) `(5/8)^(-7)xx(8/5)^(-4)`

Answer: `=(8/5)^7xx(5/8)^4`

`=(8^7)/(8^4)xx(5^4)/(5^7)`

`=(8^3)/(5^3)=(512)/(125)`

Question 7: Simplify.

(i) `(25 xx t^(-4))/(5^(-3) xx 10 xx t^(-8))`

Answer: `=(25 xx t^(-4))/(5^(-3) xx 10 xx t^(-8))`
`=(5^2 xx t^(-4))/(5^(-3) xx 5 xx 2 xx t^(-8))`
`= 5^(2+2) xx t^(-4+8) ÷ 2`
`= 5^4 xx t^4 ÷ 2`

(ii) `(3^(-5) xx 10^(-5) xx 125)/(5^(-7) xx 6^(-5))`

Answer: `= (3^(-5) xx 5^(-5) xx 2^(-5) xx 5^3)/(5^(-7) xx 6^(-5))`
`= (3^(-5) xx 5^(-5+3) xx 2^(-5))/(5^(-7) xx 6^(-5))`
`= (3^(-5) xx 5^(-2) xx 2^(-5)) /(5^(-7) xx 3^(-5) xx 2^(-5))`
`= (3^(-5+5) xx 5^(-2+7) xx 2^(-5+5))`
`= (3^0 xx 5^5 xx 2^0)`
`=3125`