Question

Simplify $\dfrac{{{3^{ - 4}} \times {{10}^{ - 5}} \times 125}}{{{5^{ - 7}} \times {6^{ - 5}}}}$.

Answer 1

Hint: We have given an expression in exponent and power form and we have to simplify it. Firstly, in the expression, we write 10 and 6 into its factor. 10 has factors 2 and 5, 6 has factors 2 and 3. Then, we apply properties on the factor of the same base. This will reduce the expression in the product of 2, 3, and 5. We apply for the property again and will find the answer.

Complete step-by-step answer:
We have given that $\dfrac{{{3^{ - 4}} \times {{10}^{ - 5}} \times 125}}{{{5^{ - 7}} \times {6^{ - 5}}}}$
We have to simplify it
We know that $10$ can be written as the product of $2$ and $5$
That is $10 = 2 \times 5$
So, ${10^{ - 5}} = {\left( {2 \times 5} \right)^{ - 5}}$
Also, we have the property that ${\left( {a \times b} \right)^n} = {a^n} \times {b^n}$
So, ${\left( {2 \times 5} \right)^{ - 5}}$ can be written as the product of ${2^{ - 5}}$ and ${5^{ - 5}}$
So, ${10^{ - 5}} = {2^{ - 5}} \times {5^{ - 5}}$
$125$ can be written as the product of three $5$ that is:
$125 = 5 \times 5 \times 5 = {5^3}$
$6$ can be written as the product of $2$ and $3$ that is:
$6 = 2 \times 3$
So, ${6^{ - 5}}$ can be written as
${6^{ - 5}} = {\left( {2 \times 3} \right)^{ - 5}}$
$ = {2^{ - 5}} \times {3^{ - 5}}$
Now we put all these values in the expression, we get:
$\dfrac{{{3^{ - 4}} \times {{10}^{ - 5}} \times 125}}{{{5^{ - 7}} \times {6^{ - 5}}}} = \dfrac{{{3^{ - 4}} \times {{10}^{ - 5}} \times {5^3}}}{{{5^{ - 7}} \times {2^{ - 5}} \times {3^{ - 5}}}}$
$ = \dfrac{{{3^{ - 4}} \times {2^{ - 5}} \times {5^{ - 5}} \times {5^3}}}{{{5^{ - 7}} \times {2^{ - 5}} \times {3^{ - 5}}}}$
Now, in the numerator we have ${5^{ - 5}} \times {5^3}$. We have the property that if the base is the same and are in the product then powers can be added.
So, ${5^{ - 5}} \times {5^3} = {5^{ - 5 + 3}} = {5^{ - 2}}$
$ \Rightarrow \dfrac{{{3^{ - 4}} \times {{10}^{ - 5}} \times 125}}{{{5^{ - 7}} \times {6^{ - 5}}}} = \dfrac{{{2^{ - 5}} \times {3^{ - 4}} \times {5^{ - 2}}}}{{{2^{ - 5}} \times {3^{ - 5}} \times {5^{ - 7}}}} = \left( {\dfrac{{{2^{ - 5}}}}{{{2^{ - 5}}}}} \right) \times \left( {\dfrac{{{3^{ - 4}}}}{{{3^{ - 5}}}}} \right) \times \left( {\dfrac{{{5^{ - 2}}}}{{{5^{ - 7}}}}} \right)$
We have the property that if bases are the same and are in the division, then powers can be subtracted.
So, $\dfrac{{{3^{ - 4}} \times {{10}^{ - 5}} \times 125}}{{{5^{ - 7}} \times {6^{ - 5}}}} = {2^{ - 5 - \left( { - 5} \right)}} \times {3^{ - 4 - \left( { - 5} \right)}} \times {5^{ - 2 - \left( { - 7} \right)}}$
$ = {2^{ - 5 + 5}} \times {3^{ - 4 + 5}} \times {5^{ - 2 + 7}}$
$ = {2^0} \times {3^1} \times {5^5}$
$ = 1 \times 3 \times 5 \times 5 \times 5 \times 5 \times 5$
$ = 9375$

So, $\dfrac{{{3^{ - 4}} \times {{10}^{ - 5}} \times 125}}{{{5^{ - 7}} \times {6^{ - 5}}}} = 9375$

Note: Power denotes the repeated multiplication of the factors and the number which is raised to the base factor is the exponent. This is the main difference between power and exponent.
For example: ${3^2}$ is the power where $3$ is the base and $2$ is the exponent.
Base Number: A base number is a number that is multiplied by itself.