Question

How do you simplify${8^{ - 1}}$?

Answer 1

Hint:
Here we must know that ${a^{ - b}}$ can be written in the form of $\dfrac{1}{{{a^b}}}$and also that ${a^b} = a.a.a.a........{\text{b times}}$ Hence by using this property we can simplify the above given term also.

Complete Step by step Solution:
Here we are given the term ${8^{ - 1}}$ which we need to simplify. So first of all we need to convert this value which in the form of the exponent and the degree into the form where we have numerator and denominator and then we can cancel those terms in the numerator and denominator with the common factor of them and hence we can further get the simplest form of the given term or the fraction.
We know that ${a^{ - b}}$ can be written in the form of $\dfrac{1}{{{a^b}}}$ and therefore we can say that ${8^{ - 1}}$ can be written as $\dfrac{1}{8}$ and therefore we get the simplified form as $\dfrac{1}{8}$
Now we can divide this term $1{\text{ by 8}}$ and multiply the term $1000$ in the numerator and denominator so that we can avoid the division with the long division method as it can take time. In this type of method, we can simply cancel the terms in the numerator and denominator and get the simplest form of the fraction.
 Now we can write $\dfrac{1}{8}{\text{ as }}\dfrac{1}{8} \times \dfrac{{1000}}{{1000}}$ as we have multiplied thousand both sides in the numerator as well as the denominator.
So we get $\dfrac{{1000}}{8} \times \dfrac{1}{{1000}}$
Now we know that $\dfrac{{1000}}{8} = \dfrac{{500}}{4} = \dfrac{{250}}{2} = 125$
So we get that $\dfrac{{1000}}{8} \times \dfrac{1}{{1000}}$$ = \dfrac{{125}}{{1000}} = 0.125$

Hence we can write the value of ${8^{ - 1}}$ as $0.125$

Note:
In these types of problems if we are given the term ${0.08^{ - 2}}$ then we must know what it's fraction will be.
We can write it as ${0.08^{ - 2}} = {\left( {\dfrac{8}{{100}}} \right)^{ - 2}} = {\left( {\dfrac{{100}}{8}} \right)^2} = \dfrac{{10000}}{{64}}$ and then we can simplify the fraction to get the simplest form of the fraction. Hence a student must know the correct way to write this form into the fraction form.