Question

How do you simplify \[-{{5}^{-3}}\]?

Answer 1

Hint: To solve the given problem we should know some of the exponential properties. The first property that we should know is how to evaluate expressions of form \[{{a}^{-b}}\]. This term is evaluated as \[\dfrac{1}{{{a}^{b}}}\]. The second exponential property, we should know states that \[{{\left( ab \right)}^{m}}={{a}^{m}}{{b}^{m}}\]. We will use these properties, to simplify the given expression.

Complete step by step solution:
We are asked to simplify the expression \[-{{5}^{-3}}\]. This expression is of the form \[{{a}^{-b}}\]. On comparing with this form, we get \[a=-5\] , and \[b=3\]. We know that expressions of these forms are evaluated as \[\dfrac{1}{{{a}^{b}}}\]. Substituting the values of the a and b, we get
\[\Rightarrow {{\left( -5 \right)}^{-3}}=\dfrac{1}{{{\left( -5 \right)}^{3}}}\]
We know the exponential property \[{{\left( ab \right)}^{m}}={{a}^{m}}{{b}^{m}}\]. For the denominator of the above expression, we have \[a=-1\], \[b=5\], and \[m=3\]. Using the above exponential property, we can express the denominator of the term \[\dfrac{1}{{{\left( -5 \right)}^{3}}}\] as follows,
\[\Rightarrow \dfrac{1}{{{\left( -5 \right)}^{3}}}=\dfrac{1}{{{\left( -1 \right)}^{3}}\times {{5}^{3}}}\]
We know that to find the cube of a number, we need to multiply it with itself three times. Thus, the cube of \[-1\] is \[-1\times -1\times -1\] which equals \[-1\]. And, the cube of 5 is \[5\times 5\times 5\] which equals 125.
Substituting these values in the above expression, we get
\[\begin{align}
  & \Rightarrow \dfrac{1}{{{\left( -1 \right)}^{3}}\times {{5}^{3}}}=\dfrac{1}{-1\times 125} \\
 & \Rightarrow -\dfrac{1}{125} \\
\end{align}\]

Note: For these types of problems, we should know some of the special exponents. We have already seen one of them in the above example, which is \[{{a}^{-b}}\] and is evaluated as \[\dfrac{1}{{{a}^{b}}}\]. We should also know how to evaluate expressions of the form \[{{a}^{\dfrac{1}{n}}}\] this expression is simplified as \[\sqrt[n]{a}\], which means the \[{{n}^{th}}\] root of the a.