Question

How do you simplify \[{{\left( \dfrac{8}{125} \right)}^{\dfrac{2}{3}}}\]?

Answer 1

Hint: To solve the given problem, we will need some of the properties of the exponent. These properties are as follows, the first property states that, \[{{\left( \dfrac{a}{b} \right)}^{n}}=\dfrac{{{a}^{n}}}{{{b}^{n}}}\]. Another property of exponents which states that \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}\].

Complete answer:
The given exponential expression is \[{{\left( \dfrac{8}{125} \right)}^{\dfrac{2}{3}}}\], we have to find its value. We will use the property of exponents which states that \[{{\left( \dfrac{a}{b} \right)}^{n}}=\dfrac{{{a}^{n}}}{{{b}^{n}}}\]. Using this property in the above expression, we get
\[\Rightarrow {{\left( \dfrac{8}{125} \right)}^{\dfrac{2}{3}}}=\dfrac{{{\left( 8 \right)}^{\dfrac{2}{3}}}}{{{\left( 125 \right)}^{\dfrac{2}{3}}}}\]
The above expression can also be written as, \[\dfrac{{{\left( 8 \right)}^{2\times \dfrac{1}{3}}}}{{{\left( 125 \right)}^{2\times \dfrac{1}{3}}}}\]. We know the property of exponents which states that \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}\], as this is a properties are true for both directions, we can also use this in reverse direction. Using the property in reverse direction for the above expression, we get
\[\Rightarrow \dfrac{{{\left( 8 \right)}^{2\times \dfrac{1}{3}}}}{{{\left( 125 \right)}^{2\times \dfrac{1}{3}}}}=\dfrac{{{\left( {{8}^{2}} \right)}^{\dfrac{1}{3}}}}{{{\left( {{125}^{2}} \right)}^{\dfrac{1}{3}}}}\]
We know that the square of 8 is 64, and the square of 125 is 15,625. Hence, we can express it as,
\[\Rightarrow {{8}^{2}}=64\And {{125}^{2}}=15,625\]
Using these values in the above expression, we get
\[\Rightarrow \dfrac{{{\left( {{8}^{2}} \right)}^{\dfrac{1}{3}}}}{{{\left( {{125}^{2}} \right)}^{\dfrac{1}{3}}}}=\dfrac{{{\left( 64 \right)}^{\dfrac{1}{3}}}}{{{\left( 15625 \right)}^{\dfrac{1}{3}}}}\]
We also know that 64 is the cube of 4, and 15,625 is the cube of 25. Hence, we can express it as,
\[\Rightarrow {{4}^{3}}=64\And {{25}^{3}}=15625\]
Taking the cube root of both sides of the above two expressions, we get
\[\Rightarrow {{\left( {{4}^{3}} \right)}^{\dfrac{1}{3}}}={{\left( 64 \right)}^{\dfrac{1}{3}}}\And {{\left( {{25}^{3}} \right)}^{\dfrac{1}{3}}}={{\left( 15625 \right)}^{\dfrac{1}{3}}}\]
This can also be expressed as, \[4={{\left( 64 \right)}^{\dfrac{1}{3}}}\And 25={{\left( 15625 \right)}^{\dfrac{1}{3}}}\]. Using these values for the calculation of \[\dfrac{{{\left( 64 \right)}^{\dfrac{1}{3}}}}{{{\left( 15625 \right)}^{\dfrac{1}{3}}}}\], we get
\[\Rightarrow \dfrac{{{\left( 64 \right)}^{\dfrac{1}{3}}}}{{{\left( 15625 \right)}^{\dfrac{1}{3}}}}=\dfrac{4}{25}\]
Hence, the value of \[{{\left( \dfrac{8}{125} \right)}^{\dfrac{2}{3}}}\] equals \[\dfrac{4}{25}\].

Note: To solve these types of problems, one should know the properties of exponents. Also values of squares, cubes, square roots, and cube roots of the numbers. It is not necessary to use the properties in the same order as done in the solution. Any property can be applied first.