Question

How do you simplify \[\dfrac{{{10}^{-2}}}{{{10}^{-4}}}\] ?

Answer 1

Hint: We know that multiplying and dividing and expression does not change its value. This property can be used here. To simplify the expression means we have to remove any radical power term from the denominator, here the term present in the denominator is of lower power than the numerator. So, to simplify it we have to multiply and divide the expression by the same term as the denominator.

Complete step by step answer:
We are asked to simplify the expression \[\dfrac{{{10}^{-2}}}{{{10}^{-4}}}\] .
Term present in the numerator is \[{{10}^{-4}}\] .
We can rationalize the denominator by multiplying with \[{{10}^{4}}\] but only multiplying at the denominator may change the value of the expression. To prevent this, we have to multiply the same term in the numerator also.
Multiply the numerator and denominator by \[{{10}^{-4}}\] , we get
\[\dfrac{{{10}^{-2}}}{{{10}^{-4}}}\times \dfrac{{{10}^{4}}}{{{10}^{4}}}\]
it can also be written as \[\dfrac{{{10}^{-2}}\times {{10}^{4}}}{{{10}^{-4}}\times {{10}^{4}}}\]
As we know the property of the exponents that,
\[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]
Using this property in the above expression, we get
\[\dfrac{{{10}^{-2}}\times {{10}^{4}}}{{{10}^{-4}}\times {{10}^{4}}}=\dfrac{{{10}^{-2+4}}}{{{10}^{-4+4}}}\]
$\Rightarrow \dfrac{{{10}^{2}}}{{{10}^{0}}}=100$

So, the simplified form of the given expression \[\dfrac{{{10}^{-2}}}{{{10}^{-4}}}\] is 100.

Note: To simplify the expression with radical power in the denominator, you need to rationalize the denominator. This can easily be done by multiplying and dividing the term that is in the denominator. You can easily confirm whether your answer is correct or not. To do this multiply the denominator term to the value we got after simplifying the expression. If it gives the value equal to the numerator our solution is correct. Here,
\[\Rightarrow {{10}^{-4}}\times 100={{10}^{-4}}\times {{10}^{2}}\]
By the property of exponents, we used the solution \[{{10}^{-4+2}}={{10}^{-2}}\] .
Which equals the numerator, so our solution is correct.