Question

Evaluate the following (ii) ​

Answer 1

Solution:

Given to evaluate:

$\tt (3^{-1}+4^{-1})^{-2}$

Can be written as:

$\tt=\bigg[\dfrac{1}{3}+\dfrac{1}{4}\bigg]^{-2}\:\:\bigg[\because x^{-1}=\dfrac{1}{x}, x\ne 0\:\bigg]$

$\tt=\bigg[\dfrac{4+3}{12}\bigg]^{-2}$

$\tt=\bigg(\dfrac{7}{12}\bigg)^{-2}$

$\tt=\bigg(\dfrac{12}{7}\bigg)^{2}$

$\tt=\dfrac{144}{49}$

Which is our required answer.

Learn More:

Laws of exponents for real numbers.

If a, b are positive real numbers and m, n are rational numbers, then the following results hold.

$\tt1. \:  \:  {a}^{m}  \times  {a}^{n}  =  {a}^{m + n}$

$\tt2. \:  \:  ({a}^{m})^{n}  =  {a}^{mn}$

$\tt3. \:  \:  \dfrac{ {a}^{m} }{ {a}^{n} }  =  {a}^{m - n}$

$\tt4. \:  \:  {a}^{m} \times  {b}^{m} =  {(ab)}^{m}$

$\tt5. \: \:   \bigg(\dfrac{a}{b} \bigg)^{m}  =  \dfrac{ {a}^{m} }{ {b}^{m} }$

$\tt6. \:  \:  {a}^{ - n} =  \dfrac{1}{ {a}^{n} }$

$\tt7. \:  \:  {a}^{n} =  {b}^{n} \rightarrow a = b, n \neq0$

$\tt8. \:  \:  {a}^{m} =  {a}^{n} \rightarrow m = n, a \neq 1$