Question

Simplify \[\dfrac{{{3^{ - 5}} \times {{10}^{ - 5}} \times 125}}{{{5^{ - 7}} \times {6^{ - 5}}}}\].
A) \[{5^5}\]
B) \[{5^4}\]
C) \[{5^2}\]
D) \[1\]

Answer 1

Hint: Here in this question, we have to express the given complex number into the simplest form. This can be solved by using some standard properties of exponents and further simplified by using a basic arithmetic operation to get the required solution which is in the simplest form.

Complete step by step solution:
The exponential number is defined as a number of times the number is multiplied by itself.
Some Properties of exponents are:
Product property: \[{a^m} \times {a^n} = {a^{m + n}}\]
Power of a power property: \[{\left( {{a^m}} \right)^n} = {a^{mn}}\]
Power of a product property: \[{\left( {ab} \right)^m} = {a^m}{b^m}\]
Negative exponent property: \[{a^{ - m}} = \dfrac{1}{{{a^m}}}\] ; \[a \ne 0\]
Zero exponent property: \[{a^0} = 1\] ; \[a \ne 0\]
Quotients of powers: \[\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}\] ; \[a \ne 0\]
Power of quotient: \[{\left( {\dfrac{a}{b}} \right)^m} = \dfrac{{{a^m}}}{{{b^m}}}\] ; \[b \ne 0\]
Now, consider the given question:
\[ \Rightarrow \,\,\,\,\dfrac{{{3^{ - 5}} \times {{10}^{ - 5}} \times 125}}{{{5^{ - 7}} \times {6^{ - 5}}}}\]
Now, by using a negative exponent property the above expression can be written as
\[ \Rightarrow \,\,\,\,\dfrac{{\dfrac{1}{{{3^5}}} \times \dfrac{1}{{{{10}^5}}} \times 125}}{{\dfrac{1}{{{5^7}}} \times \dfrac{1}{{{6^5}}}}}\]
Here, \[{5^7}\] can be written as: \[25 \times {5^5}\], then the above equation becomes
\[ \Rightarrow \,\,\,\,\dfrac{{\dfrac{1}{{{3^5}}} \times \dfrac{1}{{{{10}^5}}} \times 125}}{{\dfrac{1}{{25 \times {5^5}}} \times \dfrac{1}{{{6^5}}}}}\]
Now, by using a Power of a product property of exponent the above expression can be written as
\[ \Rightarrow \,\,\,\,\dfrac{{\dfrac{{125}}{{{{\left( {3 \times 10} \right)}^5}}}}}{{\dfrac{1}{{25 \times {{\left( {5 \times 6} \right)}^5}}}}}\]
on multiplication, we have
\[ \Rightarrow \,\,\,\,\dfrac{{\dfrac{{125}}{{{{\left( {30} \right)}^5}}}}}{{\dfrac{1}{{25{{\left( {30} \right)}^5}}}}}\]
Or it can be written as
\[ \Rightarrow \,\,\,\dfrac{{125}}{{{{\left( {30} \right)}^5}}} \times 25{\left( {30} \right)^5}\]
On cancelling, the like terms we have
\[ \Rightarrow \,\,\,125 \times 25\]
As we know, 125 is the cubic number of 5 i.e., \[125 = {5^3}\] and 25 is the square number of 5 i.e., \[25 = {5^2}\], then
\[ \Rightarrow \,\,\,{5^3} \times {5^2}\]
Now by the product property of exponent, we can written as
\[ \Rightarrow \,\,\,{5^{3 + }}^2\]
\[ \Rightarrow \,\,\,{5^5}\]
Hence, the simplest form of \[\dfrac{{{3^{ - 5}} \times {{10}^{ - 5}} \times 125}}{{{5^{ - 7}} \times {6^{ - 5}}}}\] is \[{5^5}\].

Note:
The exponential number is the number which is the base number and exponent number. If a number has an exponent, which means the base number is multiplied, the exponent number times. If a base number has a negative exponent number then it will be written in the form of fraction. While cancelling the terms you should know the multiplication of tables.