Question

How do you simplify \[{\left( {{x^{\dfrac{2}{3}}}.{y^{ - \dfrac{3}{4}}}} \right)^{\dfrac{{12}}{3}}}\] ?

Answer 1

Hint: Here we have a simple algebraic expression. We can simplify this using the laws of indices. Without knowing the laws of indices it is impossible to simplify any mathematical equation or expression. To solve this we use the third law of indices (brackets). That is \[ \Rightarrow {\left( {{x^m}} \right)^n} = {x^{m \times n}}\] .

Complete step-by-step answer:
Given,
 \[{\left( {{x^{\dfrac{2}{3}}}.{y^{ - \dfrac{3}{4}}}} \right)^{\dfrac{{12}}{3}}}\] .
As we have \[\dfrac{{12}}{3}\] as a indices of the whole expression we can simplify that,
 \[\dfrac{{12}}{3} = 4\] .
 \[{\left( {{x^{\dfrac{2}{3}}}.{y^{ - \dfrac{3}{4}}}} \right)^4}\]
Now applying the law if brackets we have,
 \[ \Rightarrow {\left( {{x^{\dfrac{2}{3}}}.{y^{ - \dfrac{3}{4}}}} \right)^4} = {\left( {{x^{\dfrac{2}{3}}}} \right)^4}.{\left( {{y^{ - \dfrac{3}{4}}}} \right)^4}\]
 \[ \Rightarrow {\left( {{x^{\dfrac{2}{3}}}.{y^{ - \dfrac{3}{4}}}} \right)^4} = {\left( {{x^{\dfrac{2}{3}}}} \right)^4}.{\left( {{y^{ - \dfrac{3}{4}}}} \right)^4}\]
 \[ = {\left( {{x^{\dfrac{2}{3}}}} \right)^4}.{\left( {{y^{ - \dfrac{3}{4}}}} \right)^4}\]
 \[ = \left( {{x^{\dfrac{2}{3} \times 4}}} \right).\left( {{y^{ - \dfrac{3}{4} \times 4}}} \right)\]
 \[ = \left( {{x^{\dfrac{8}{3}}}} \right).\left( {{y^{ - 3}}} \right)\] . This is the simplified form.

Note: We have several laws of indices.
 \[ \bullet \] The first law: multiplication: if the two terms have the same base and are to be multiplied together their indices are added. That is \[ \Rightarrow {x^m} \times {x^n} = {x^{m + n}}\]
 \[ \bullet \] The second law: division: If the two terms have the same base and are to be divided their indices are subtracted. That is \[ \Rightarrow \dfrac{{{x^m}}}{{{x^n}}} = {x^{m - n}}\]
 \[ \bullet \] The third law: brackets: If a term with a power is itself raised to a power then the powers are multiplied together. That is \[ \Rightarrow {\left( {{x^m}} \right)^n} = {x^{m \times n}}\]
 \[ \bullet \] As we have the second law of indices which helps to explain why anything to the power of zero is equal to one. \[ \Rightarrow {x^0} = 1\]
 \[ \bullet \] Negative power \[ \Rightarrow {x^{ - n}} = \dfrac{1}{{{x^m}}}\]
 \[ \bullet \] The fractional power \[ \Rightarrow {x^{\dfrac{m}{n}}} = \left( {\sqrt[n] {m}} \right)\] . We use them depending on the given expression.