Question

How do you simplify $\dfrac{{{{( - 10)}^5}}}{{{{( - 10)}^9}}}$

Answer 1

Hint: Here we will use the rules of exponents and multiplication to expand the sum and then we will simplify it. Finally we get the required answer.

Complete step-by-step solution:
The given question is: $ \Rightarrow \dfrac{{{{( - 10)}^5}}}{{{{( - 10)}^9}}}$
Now since the numerator and denominator both are in the form of exponents and have powers, we can expand and write the equation.
Now since the numerator has a power of $5$ and the denominator has a power of $9$ we can expand and write:
 $ \Rightarrow \dfrac{{ - 10 \times - 10 \times - 10 \times - 10 \times - 10}}{{ - 10 \times - 10 \times - 10 \times - 10 \times - 10 \times - 10 \times - 10 \times - 10 \times - 10}}$
Now since the same term is in the numerator and denominator, we can cancel the terms as:
$ \Rightarrow \dfrac{1}{{ - 10 \times - 10 \times - 10 \times - 10}}$
Now on simplifying the denominator we get:
$ \Rightarrow \dfrac{1}{{10000}}$
Now this can be written in the exponent format since ${10^4} = 10000$.
Therefore, on substituting we get:
$ \Rightarrow \dfrac{1}{{{{10}^4}}}$

$\dfrac{1}{{{{10}^4}}}$ is the required answer.

Note: In the given question we have expanded the given equation and cancelled the terms. This could be directly done using the rule of exponents because the coefficient is the same in both the numerator and the denominator. Therefore:
$ \Rightarrow \dfrac{{{{( - 10)}^5}}}{{{{( - 10)}^9}}} = \dfrac{1}{{{{( - 10)}^4}}}$
Since the value over here is $\dfrac{1}{{{{( - 10)}^4}}}$ rather than $\dfrac{1}{{{{10}^4}}}$ it doesn’t make a difference because the value is same for both.
Exponents are used to represent large numbers in a small manner by assigning those powers. To simplify the calculation of exponential numbers, logarithm is used.
Logarithm is a method by which numbers which are in multiplication or division can be simplified using addition or subtraction respectively.
It is to be remembered that in multiplication when even numbers of positive or even number of negative terms are multiplied the answer will be positive and the answer will be negative if an odd number of negative numbers are multiplied.