Question

How do you simplify \[{1000^{\dfrac{{ - 2}}{3}}}\] ?

Answer 1

Hint: We can see this problem is from indices and powers. This number is given as having 1000 as base and \[\dfrac{{ - 2}}{3}\] as power. But since we have to simplify this we can write \[\dfrac{{ - 2}}{3}\] as a cube-root also but in the denominator of the fraction. Then the same number \[{1000^{\dfrac{{ - 2}}{3}}}\] is written with positive power and then we will find it’s cube-root. That will be our answer.

Complete step-by-step answer:
Given that \[{1000^{\dfrac{{ - 2}}{3}}}\]
This is of the form \[{a^{ - n}}\] . But we can rewrite it using the laws of indices and powers as \[ \Rightarrow {a^{ - n}} = \dfrac{1}{{{a^n}}}\]
Thus we will apply this same on the question above.
 \[ \Rightarrow {1000^{\dfrac{{ - 2}}{3}}} = \dfrac{1}{{{{1000}^{\dfrac{2}{3}}}}}\]
Now this power is nothing but the cube-root of that number in base.
 \[ \Rightarrow {1000^{\dfrac{{ - 2}}{3}}} = \dfrac{1}{{{{\left( {{{\left( {1000} \right)}^{\dfrac{1}{3}}}} \right)}^2}}}\]
Now to the power \[\dfrac{1}{3}\] is the cube-root
 \[ \Rightarrow {1000^{\dfrac{{ - 2}}{3}}} = \dfrac{1}{{{{\left( {\sqrt[3] {{1000}}} \right)}^2}}}\]
We know that 1000 is the cube-root of 10. So we can write this as,
 \[ \Rightarrow {1000^{\dfrac{{ - 2}}{3}}} = \dfrac{1}{{{{\left( {10} \right)}^2}}}\]
Square of 10 is 100.
 \[ \Rightarrow {1000^{\dfrac{{ - 2}}{3}}} = \dfrac{1}{{100}}\]
This is our simplified answer.
So, the correct answer is “$\dfrac{1}{{100}}$”.

Note: If students get confused with how to concert that fraction written in place of power as which root then they can go for the alternative method mentioned below.
 \[ \Rightarrow {1000^{\dfrac{{ - 2}}{3}}} = \dfrac{1}{{{{1000}^{\dfrac{2}{3}}}}}\]
Now we know that 1000 is the cube of 10. So we can write as,
 \[ \Rightarrow {1000^{\dfrac{{ - 2}}{3}}} = \dfrac{1}{{{{\left( {{{\left( {10} \right)}^3}} \right)}^{\dfrac{2}{3}}}}}\]
Now multiplying the powers we get,
 \[ \Rightarrow {1000^{\dfrac{{ - 2}}{3}}} = \dfrac{1}{{{{\left( {10} \right)}^2}}}\]
Now square of 10 is 100 as we know,
 \[ \Rightarrow {1000^{\dfrac{{ - 2}}{3}}} = \dfrac{1}{{100}}\]
This is our answer.
These rules or laws of indices help us to minimize the problems and get the answer in very less time. These powers can be positive and negative but can be molded according to our convenience while solving the problem. Also note that cube-root, square-root is fractions with 1 as numerator and respective root in denominator but the power in our problem is first cube root then square. Hope this will help you!