Question

How do you simplify \[{\left( { - \dfrac{{27}}{{125}}} \right)^{ - \dfrac{2}{3}}}\]?

Answer 1

Hint: Here in this question we have to simplify the given number. The number is a fractional exponential number. By using the simple arithmetic operations and the definition and properties of the exponent, we can determine the value or we can simplify it.

Complete step-by-step answer:
Now consider the given number \[{\left( { - \dfrac{{27}}{{125}}} \right)^{ - \dfrac{2}{3}}}\]
Interchange the value of numerator and denominator, i.e., the value of the numerator is placed in the denominator. The value of the denominator is placed in the numerator. Hence the negative power will be changed into the positive. So the above number is written as
\[ \Rightarrow {\left( { - \dfrac{{125}}{{27}}} \right)^{\dfrac{2}{3}}}\]
The exponent number is written as
\[ \Rightarrow {\left( {{{\left( { - \dfrac{{125}}{{27}}} \right)}^{\dfrac{1}{3}}}} \right)^2}\]
Let we write the negative sign separately,
\[ \Rightarrow {\left( {{{\left( { - 1} \right)}^{\dfrac{1}{3}}}} \right)^2}{\left( {{{\left( {\dfrac{{125}}{{27}}} \right)}^{\dfrac{1}{3}}}} \right)^2}\]
As we know that the exponent \[\dfrac{1}{3}\] represents the cubic root. Therefore the cubic root of 125 is 5 i.e., \[125 = 5 \times 5 \times 5\]and the cubic root of 27 is 3 i.e., \[27 = 3 \times 3 \times 3\]. The cubic root of (-1) is -1. i.e., \[ - 1 = - 1 \times - 1 \times - 1\], On considering above inequalities the given number can be written as
\[ \Rightarrow {\left( { - 1} \right)^2}{\left( {\dfrac{5}{3}} \right)^2}\]
As we know that the exponent \[2\] represents the square. Therefore the square of 5 is 25 i.e., \[{5^2} = 5 \times 5 = 25\]and the square of 3 is 9 i.e., \[{3^2} = 3 \times 3 = 9\]. The square of (-1) is 1. i.e., \[{\left( { - 1} \right)^2} = - 1 \times - 1 = 1\], On considering above inequalities the given number can be written as
\[ \Rightarrow \dfrac{{25}}{9}\]
There is no common number for the number 25 and 9. On further simplification we get the value in terms of decimal. so we keep it as it is.
Hence we have simplified the given numeral or number.

Note: The exponential number is a number obtained by multiplying the number by itself by number of times. While solving exponential numbers we must know about the law of exponents and law of indices. On simplifying the number, the table of multiplication is needed.