Question

Evaluate this following questions ​

Answer 1

Explanation:

Answer 2

`(i) \quad\;\;\; \left(3^0 + 4^(-1)\right) * 2^2=5`

`(ii) \quad \;\;\left(2^(-1) * 4^(-1)\right) / 2^(-2)=(1)/(2)`

`(iii) \quad \left((1)/(2)\right)^(-2)+\left((1)/(3)\right)^(-2)+\left((1)/(4)\right)^(-2)=29`

`(iv) \quad \;\: \left(3^(-1)+4^(-1)+5^(-1)\right)^0=1`

`(v) \quad \;\;\left\{\left((-2)/(3)\right)^(-2)\right \}^2=(81)/(16)`

Explanation:

To evaluate the given expressions, we can use the following exponent rules:

`\boxed{\begin{minipage}{6cm}\underline{Exponent Rules}\\\\$a^0=1$ \qquad \qquad \qquad \qquad $1^n=1$\\\\\\$a^b * a^c=a^(b+c)$ \qquad \;\;$a^b / a^c=a^(b-c)$\\\\\\$\left((a)/(b)\right)^c=(a^c)/(b^c)$ \qquad \qquad $\left((a)/(b)\right)^(-c)=\left((b)/(a)\right)^(c)$\\\\\\$a^(-n)=(1)/(a^n)$\qquad \qquad \qquad$(1)/(a^(-n))=a^n$\\\\\\$(a^b)^c=a^(bc)$\\\end{minipage}}`

Part (i)

`\left(3^0 + 4^(-1)\right) * 2^2`

`=\left(1 + (2^2)^(-1)\right) * 2^2`

`=\left(1 + 2^(-2)\right) * 2^2`

`=2^2 + 2^2* 2^(-2)`

`=2^2 + 2^(2-2)`

`=2^2 + 2^(0)`

`=4+1`

`=5`

Part (ii)

`\left(2^(-1) * 4^(-1)\right) / 2^(-2)`

`=\left(2^(-1) * (2^2)^(-1)\right) / 2^(-2)`

`= \left(2^(-1) * 2^(-2)\right) / 2^(-2)`

`=\left(2^(-1-2)\right) / 2^(-2)`

`=2^(-3) / 2^(-2)`

`=2^(-3-(-2))`

`=2^(-3+2)`

`=2^(-1)`

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

`=(1)/(2)`

Part (iii)

`\left((1)/(2)\right)^(-2)+\left((1)/(3)\right)^(-2)+\left((1)/(4)\right)^(-2)`

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

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

`=2^2+3^2+4^2`

`=4+9+16`

`=13+16`

`=29`

Part (iv)

`\left(3^(-1)+4^(-1)+5^(-1)\right)^0`

`=1`

Part (v)

`\left\{\left((-2)/(3)\right)^(-2)\right \}^2`

`=\left((-2)/(3)\right)^(-2* 2)`

`=\left((-2)/(3)\right)^(-4)`

`=\left((3)/(-2)\right)^(4)`

`=(3^4)/((-2)^4)`

`=(81)/(16)`