Question

By what number should ${\left( {\dfrac{5}{3}} \right)^{ - 1}}$be divided so that quotient may be equal to ${\left( {\dfrac{7}{3}} \right)^{ - 1}}?$

Answer 1

Hint: Here we will use the division algorithm to frame the mathematical expression and then will simplify for the required value using the law of negative exponent rule for the fractions. In fractions the numerators and the denominator are swapped when negative power is changed to the positive power.

Complete answer: Here given that dividend is $ = {\left( {\dfrac{5}{3}} \right)^{ - 1}}$
And Quotient is $ = {\left( {\dfrac{7}{3}} \right)^{ - 1}}$
So, by using the division algorithm –
Dividend = Quotient (Divisor) $ + $remainder
Here we assume that remainder is equal to zero and place the other values –
${\left( {\dfrac{5}{3}} \right)^{ - 1}} = = {\left( {\dfrac{7}{3}} \right)^{ - 1}} \times $divisor
Simplify the above expression using the negative exponent rule, here we have fraction to the negative power so simply swap numerator and the denominator with each other to convert negative power to the positive power.
${\left( {\dfrac{3}{5}} \right)^1} = = {\left( {\dfrac{3}{7}} \right)^1} \times $divisor
When power is applied to the whole bracket, it can be applied to the numerator and the denominator.
$\left( {\dfrac{{{3^1}}}{{{5^1}}}} \right) = = \left( {\dfrac{{{3^1}}}{{{7^1}}}} \right) \times $Divisor
Make the required term as the subject and perform cross multiplication –
Divisor $ = \left( {\dfrac{3}{5}} \right) \times \left( {\dfrac{7}{3}} \right)$
Common terms from the numerator and the denominator cancel each other.
Divisor $ = \left( {\dfrac{7}{5}} \right)$
Hence the ${\left( {\dfrac{5}{3}} \right)^{ - 2}}$ should be divided by $\left( {\dfrac{7}{5}} \right)$ or ${\left( {\dfrac{5}{7}} \right)^{ - 1}}$

Note:
Always remember the division algorithm properly which can be stated as that the dividend is equal to the sum of the remainder with the product of the divisor and the quotient. Be good in multiples and the division and apply it accordingly. Always cross-check the values by using the division algorithm. Be good in different laws for the powers and exponents and apply wisely.