Question

How do you simplify \[{{\left( 182 \right)}^{-3}}\] ?

Answer 1

Hint: The easiest way to solve this problem is to first express the given expression as a fraction raised to some power. We then evaluate the fraction. Having done this, we arrive at the simplified form of the given expression.

Complete step by step answer:
Any power form of a number can be represented as
\[{{a}^{b}}\]
Where, ‘a’ is called the “base” and ‘b’ is called the “index” or “power” or “exponent”.
In the given problem, let \[x\] be \[{{(182)}^{-3}}\] .
Therefore, \[x\] can be written as,
\[x=\dfrac{1}{{{\left( 182 \right)}^{3}}}\]
What we need to find as the answer is the value of \[x\]. We further write it as,
\[x=\dfrac{1}{182\times 182\times 182}\]
Now, we need to evaluate the value of \[182\times 182\times 182\] . We get its value to be as,
\[182\times 182\times 182=6,028,568\] . Now as per the question, we need to find out the reciprocal of this value. Till now what we have got is,
\[x=\dfrac{1}{6028568}\]
Since, \[6028568\] , is a very large number, we multiply \[{{10}^{7}}\] ( because the number has 7 digits) on both the numerator and denominator, and we get,
\[\begin{align}
  & x=\dfrac{{{10}^{7}}}{6028568\times {{10}^{7}}} \\
 & \Rightarrow x=\dfrac{10000000}{6028568}\times {{10}^{-7}} \\
\end{align}\]
Now, finding the value of \[\dfrac{10000000}{6028568}\]we get,
\[\dfrac{10000000}{6028568}=1.6588\]

Thus putting the above value in the original equation we get,
\[x=1.6588\times {{10}^{-7}}\].

Note: The most important point that we need to keep in mind while solving square or cube or any other power problems is that, we must be very careful with conversion of negative powers to positive powers and must always remember that the \[{{a}^{-n}}^{th}\] power is equivalent to \[\dfrac{1}{{{a}^{n}}}^{th}\] power. We must also be careful with the properties, and should not overlap one property with the other.