Question

How do you simplify ${{\left( -8 \right)}^{-3}}$ ?

Answer 1

Hint: For getting the simplified value of the given question ${{\left( -8 \right)}^{-3}}$ . We will have to understand the form of power. Here, the given question is in the form of ${{\left( n \right)}^{-m}}$. So, we will make the power of the question into positive term like ${{\left( n \right)}^{-m}}$ to ${{\left( \dfrac{1}{n} \right)}^{m}}$. Here, ${{\left( \dfrac{1}{n} \right)}^{m}}$ defines that the value $\left( \dfrac{1}{n} \right)$ repeatedly multiplication of itself $m$ times as $\left( \dfrac{1}{n}\times \dfrac{1}{n}\times \dfrac{1}{n}\times ...m\text{ times} \right)$. After multiplication we will find the simplified value of the given question ${{\left( -8 \right)}^{-3}}$ .

Complete step by step solution:
Since, the given question is ${{\left( -8 \right)}^{-3}}$ and it is in the form of ${{\left( n \right)}^{-m}}$. So, we will make the power term from negative to positive as:
$\Rightarrow {{\left( n \right)}^{-m}}={{\left( \dfrac{1}{n} \right)}^{m}}$
Here, we will replace the above term with the given question. So, we will write the given question as:
$\Rightarrow {{\left( -8 \right)}^{-3}}={{\left( -\dfrac{1}{8} \right)}^{3}}$
Now, the power defines the total number of the multiplication of the number with itself. We can understand it as:
$\Rightarrow {{\left( \dfrac{1}{n} \right)}^{m}}=\left( \dfrac{1}{n}\times \dfrac{1}{n}\times \dfrac{1}{n}\times ...m\text{ times} \right)$
Here, $\left( \dfrac{1}{n} \right)$multiplied by $\left( \dfrac{1}{n} \right)$ $m$ times. So, we can use this method to the given question after getting the positive form of the given question as:
$\Rightarrow {{\left( -\dfrac{1}{8} \right)}^{3}}=\left( -\dfrac{1}{8} \right)\times \left( -\dfrac{1}{8} \right)\times \left( -\dfrac{1}{8} \right)$
The above form the question shows that we will have to multiply them. Since, the multiplication of $1$ will be always $1$ if we multiply infinite times it to itself. So we can write the above equation as:
$\Rightarrow {{\left( -\dfrac{1}{8} \right)}^{3}}=\dfrac{1}{\left( -8 \right)\times \left( -8 \right)\times \left( -8 \right)}$
 And for $8$ , we will use the table of $8$ respectively as:
\[\Rightarrow {{\left( -\dfrac{1}{8} \right)}^{3}}=\dfrac{1}{\left[ \left( -8 \right)\times \left( -8 \right) \right]\times \left( -8 \right)}\]
Since, we will get $64$ after multiplying $8$ by $8$ and multiplication of two negative terms shall be positive as:
$\Rightarrow {{\left( -\dfrac{1}{8} \right)}^{3}}=\dfrac{1}{64\times \left( -8 \right)}$
Now, we will multiply $64$ and $8$ with each other. We will have $512$ after multiplying $64$ by $8$and multiplication of one negative term with one positive term results negative term as:
$\Rightarrow {{\left( -\dfrac{1}{8} \right)}^{3}}=\left( -\dfrac{1}{512} \right)$
Since, the above value in fraction, we will convert it into decimal as:
$\Rightarrow \dfrac{1}{512}=1\div 512\simeq 0.001953$
Since, the whole term is negative so the result will be negative also as:
$\Rightarrow \left( -\dfrac{1}{512} \right)\simeq \left( -0.001953 \right)$
Hence, the simplified value of the given question ${{\left( -8 \right)}^{-3}}$is $\left( -\dfrac{1}{512} \right)$ as a fraction number and is approximately equal to $\left( -0.001953 \right)$ as a decimal number.

Note: To get the simplified value of the given question ${{\left( -8 \right)}^{-3}}$ , we will write it as ${{\left( -8 \right)}^{3\times \left( -1 \right)}}$. Now, we can also write it as ${{\left[ {{\left( -8 \right)}^{3}} \right]}^{\left( -1 \right)}}$ . Here, we will expand the term as ${{\left[ \left( -8 \right)\times \left( -8 \right)\times \left( -8 \right) \right]}^{-1}}$ . Since, the term is in multiplication form, we will multiply them step by step as:
$\begin{align}
  & \Rightarrow {{\left[ \left( -8 \right)\times \left( -8 \right)\times \left( -8 \right) \right]}^{-1}} \\
 & \Rightarrow {{\left[ 64\times \left( -8 \right) \right]}^{-1}} \\
 & \Rightarrow {{\left[ -512 \right]}^{-1}} \\
\end{align}$
Since, power of the term is still negative. So, we will change it into positive as:
$\Rightarrow {{\left[ -512 \right]}^{-1}}=\left[ -\dfrac{1}{512} \right]$
Now, we will convert the above term from fraction to decimal number as:
$\Rightarrow \left( -\dfrac{1}{512} \right)\simeq \left( -0.001953 \right)$
Hence, the above solution is correct.