Question

How do you simplify \[\dfrac{{{5}^{8}}}{{{5}^{5}}}\]?

Answer 1

Hint: This question belongs to the topic of logarithms. In solving this question, we will first suppose the term \[\dfrac{{{5}^{8}}}{{{5}^{5}}}\] as x. After that, we will equate them. After that, we will take log to both sides of the equation. After using some formulas of logarithms, we will solve and find the value of x. After that, we will get our answer. After that, we will see the alternate method to solve this question.

Complete step by step solution:
Let us solve this question.
In this question, we have asked to simplify \[\dfrac{{{5}^{8}}}{{{5}^{5}}}\].
Let the term \[\dfrac{{{5}^{8}}}{{{5}^{5}}}\] be equal to x. Then, we can write
\[x=\dfrac{{{5}^{8}}}{{{5}^{5}}}\]
Taking ‘log’ to both the side of equation, we can write the above equation as
\[\Rightarrow \log x=\log \left( \dfrac{{{5}^{8}}}{{{5}^{5}}} \right)\]
Now, using the formula \[\log \left( \dfrac{a}{b} \right)=\log a-\log b\], we can write the above equation as
\[\Rightarrow \log x=\log {{5}^{8}}-\log {{5}^{5}}\]
Now, using the formula \[\log {{a}^{n}}=n\log a\], we can write the above equation as
\[\Rightarrow \log x=8\log 5-5\log 5\]
We can write the above equation as
\[\Rightarrow \log x=8\left( \log 5 \right)-5\left( \log 5 \right)\]
The above equation can also be written as
\[\Rightarrow \log x=\left( 8-5 \right)\log 5\]
The above equation can also be written as
\[\Rightarrow \log x=3\log 5\]
Using the formula \[\log {{a}^{n}}=n\log a\], we can write the above equation as
\[\Rightarrow \log x=\log {{5}^{3}}\]
Now, removing ‘log’ from the both side of equation, we get
\[\Rightarrow x={{5}^{3}}\]
As we know that, cube of 5 is 125, so we can write
\[\Rightarrow x=125\]

Initially, we had taken x as \[\dfrac{{{5}^{8}}}{{{5}^{5}}}\] and we have found the value of x as 125. So, we can say that the simplified value of the term \[\dfrac{{{5}^{8}}}{{{5}^{5}}}\] is 125.

Note: We should have a better knowledge in the topic of logarithms. For solving this type of question, we should know the following formulas:
\[\log {{a}^{n}}=n\log a\]
\[\log \left( \dfrac{a}{b} \right)=\log a-\log b\]
We can solve this question by an alternate method.
The term we have to solve is
\[\dfrac{{{5}^{8}}}{{{5}^{5}}}\]
This can also be written as
\[\Rightarrow \dfrac{{{5}^{8}}}{{{5}^{5}}}={{5}^{8}}\times \dfrac{1}{{{5}^{5}}}\]
Using the formula \[\dfrac{1}{{{a}^{n}}}={{a}^{-n}}\], we can write the above term as
\[\Rightarrow \dfrac{{{5}^{8}}}{{{5}^{5}}}={{5}^{8}}\times {{5}^{-5}}\]
Using the formula \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\], we can write the above term as
\[\Rightarrow \dfrac{{{5}^{8}}}{{{5}^{5}}}={{5}^{8-5}}={{5}^{3}}\]
As we know that the cube of 5 is 125.
So, we get that the simplified value of the term \[\dfrac{{{5}^{8}}}{{{5}^{5}}}\] is 125.