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${{\left( \dfrac{-5}{6} \right)}^{\dfrac{3}{4}}}\div {{\left( \dfrac{-5}{6} \right)}^{\dfrac{7}{6}}}$

Last updated date: 16th Jun 2024
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Hint: To divide the exponents or the powers with the same base or the same term, simply subtract the powers. As, the division is just the opposite of the multiplication, so when you add the powers in the multiplication, just subtract the powers in case of the division with the same base. For example \[{{2}^{5}}\div {{2}^{2}}={{2}^{5-2}}={{2}^{3}}\]

Complete step-by-step answer:
Here, by using the property – in case of division of the powers with the same base, simply subtracting the powers
$\Rightarrow {{\left( \dfrac{-5}{6} \right)}^{\dfrac{3}{4}}}\div {{\left( \dfrac{-5}{6} \right)}^{\dfrac{7}{6}}}={{\left( \dfrac{-5}{6} \right)}^{\dfrac{3}{4}-\dfrac{7}{6}}}$
Take LCM of the powers on the right hand side of the equation
$\Rightarrow {{\left( \dfrac{-5}{6} \right)}^{\dfrac{3}{4}}}\div {{\left( \dfrac{-5}{6} \right)}^{\dfrac{7}{6}}}={{\left( \dfrac{-5}{6} \right)}^{\dfrac{18}{24}-\dfrac{28}{24}}}$
Now, simply the power on the right hand side of the equation. Using the identity of minus and plus, do minus and sign of greater value.
$\Rightarrow {{\left( \dfrac{-5}{6} \right)}^{\dfrac{3}{4}}}\div {{\left( \dfrac{-5}{6} \right)}^{\dfrac{7}{6}}}={{\left( \dfrac{-5}{6} \right)}^{\dfrac{-10}{24}}}$
Taking “two common” from the numerator and denominator of the power on RHS
$\Rightarrow {{\left( \dfrac{-5}{6} \right)}^{\dfrac{3}{4}}}\div {{\left( \dfrac{-5}{6} \right)}^{\dfrac{7}{6}}}={{\left( \dfrac{-5}{6} \right)}^{\dfrac{-5}{12}}}$
Therefore, the required solution is –
$\Rightarrow {{\left( \dfrac{-5}{6} \right)}^{\dfrac{3}{4}}}\div {{\left( \dfrac{-5}{6} \right)}^{\dfrac{7}{6}}}={{\left( \dfrac{-5}{6} \right)}^{\dfrac{-5}{12}}}$

Note: Always remember all the rules of multiplication and division of the fractions. Remember two basic rules - Multiply the given terms with the exponents using the general rule: ${{y}^{a}}\times {{y}^{b}}={{y}^{a+b}}$ and similarly the divide terms with the exponents using the rule: ${{y}^{a}}\div {{y}^{b}}={{y}^{a-b}}$. Do simplification carefully. Rest goes perfect in these types of questions.