Question

Simplify: \[\dfrac{{{3}^{5}}\times {{10}^{5}}\times {{2}^{5}}}{{{5}^{7}}\times {{6}^{5}}}\]

Answer 1

Hint: In this type of question we have to use the concept of exponents and powers. When we have to simplify the given expression, first we express 10 and 6 into its factor. Then we apply properties on the factors of the same base i.e. \[{{\left( a\times b \right)}^{n}}={{a}^{n}}\times {{b}^{n}}\]. Then the entire expression gets reduced in the product of 2, 3 and 5. Then again we use the properties of and will find the answer.

Complete step by step solution:
Now, we have to simplify \[\dfrac{{{3}^{5}}\times {{10}^{5}}\times {{2}^{5}}}{{{5}^{7}}\times {{6}^{5}}}\]
For this we first express 10 and 6 into its factors. 10 can be written as the product of 2 and 5 where 6 can be written as the product of 2 and 3. Hence we can write,
\[\Rightarrow 10=2\times 5\text{ and }6=2\times 3\]
Now by using the property \[{{\left( a\times b \right)}^{n}}={{a}^{n}}\times {{b}^{n}}\] we can express \[{{10}^{5}}\] and \[{{6}^{5}}\] as
\[\begin{align}
  & \Rightarrow {{10}^{5}}={{\left( 2\times 5 \right)}^{5}}={{2}^{5}}\times {{5}^{5}} \\
 & \Rightarrow {{6}^{5}}={{\left( 2\times 3 \right)}^{5}}={{2}^{5}}\times {{3}^{5}} \\
\end{align}\]
Now we put all these values in the expression, we get:
\[\Rightarrow \dfrac{{{3}^{5}}\times {{10}^{5}}\times {{2}^{5}}}{{{5}^{7}}\times {{6}^{5}}}=\dfrac{{{3}^{5}}\times {{2}^{5}}\times {{5}^{5}}\times {{2}^{5}}}{{{5}^{7}}\times {{2}^{5}}\times {{3}^{5}}}\]
As we know that if the base is same and are in product then powers can be added i.e. \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]. So by using this property in numerator we can write, \[{{2}^{5}}\times {{2}^{5}}={{2}^{5+5}}={{2}^{10}}\]
\[\Rightarrow \dfrac{{{3}^{5}}\times {{10}^{5}}\times {{2}^{5}}}{{{5}^{7}}\times {{6}^{5}}}=\dfrac{{{3}^{5}}\times {{2}^{10}}\times {{5}^{5}}}{{{5}^{7}}\times {{2}^{5}}\times {{3}^{5}}}\]
Now separating the terms with common base we can write,
\[\Rightarrow \dfrac{{{3}^{5}}\times {{10}^{5}}\times {{2}^{5}}}{{{5}^{7}}\times {{6}^{5}}}=\dfrac{{{3}^{5}}}{{{3}^{5}}}\times \dfrac{{{2}^{10}}}{{{2}^{5}}}\times \dfrac{{{5}^{5}}}{{{5}^{7}}}\]
As we know that if the base is same and are in product then powers can be added i.e. \[\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}\]
\[\Rightarrow \dfrac{{{3}^{5}}\times {{10}^{5}}\times {{2}^{5}}}{{{5}^{7}}\times {{6}^{5}}}={{3}^{5-5}}\times {{2}^{10-5}}\times {{5}^{5-7}}\]
\[\Rightarrow \dfrac{{{3}^{5}}\times {{10}^{5}}\times {{2}^{5}}}{{{5}^{7}}\times {{6}^{5}}}={{3}^{0}}\times {{2}^{5}}\times {{5}^{-2}}\]
We have \[{{a}^{-m}}=\dfrac{1}{{{a}^{m}}}\]
\[\Rightarrow \dfrac{{{3}^{5}}\times {{10}^{5}}\times {{2}^{5}}}{{{5}^{7}}\times {{6}^{5}}}=\dfrac{1\times 2\times 2\times 2\times 2\times 2}{5\times 5}\]
\[\Rightarrow \dfrac{{{3}^{5}}\times {{10}^{5}}\times {{2}^{5}}}{{{5}^{7}}\times {{6}^{5}}}=\dfrac{32}{25}\]
Hence, on simplification we get, \[\dfrac{{{3}^{5}}\times {{10}^{5}}\times {{2}^{5}}}{{{5}^{7}}\times {{6}^{5}}}=\dfrac{32}{25}\].

Note: In this type of question students have to observe the given expression first and then by using different properties of powers and exponents have to simplify the expression. Also students have to note that anything raised to 0 is always equal to 1 i.e. \[{{a}^{0}}=1\].