Question

Simplify and express the answer in exponential form:
 \[\dfrac{{[{{(4)}^5} \times {{(4)}^2}] }}{{({2^6} \times {2^4})}}\]

Answer 1

Hint: In order to simplify and write the exponential form of a given question, we have to first write any composite number as a product of prime numbers. For example - \[26 = {13^1} \times {2^1}\] . We can solve the sum by converting the numbers in prime factors and then using laws of exponent to arrive at a solution.

Complete step-by-step answer:
Exponential form is a way of writing a number as a base with the number of repeats written as a small number to its upper right to show repeated multiplications of the same number. There are two parts in a number expressed in exponential form: a base and an exponent. Example - \[{2^6}\] has \[2\] as base and \[6\] as exponent/power/index/order.
Exponential form is a shortcut method of writing repeated multiplication by same number. For example - \[64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = {2^6}\] .
Now we can proceed to simplify \[\dfrac{{[{{(4)}^5} \times {{(4)}^2}] }}{{({2^6} \times {2^4})}}\] as follows:
Converting the numbers in the prime factors, we get,
 \[ \Rightarrow \dfrac{{[{{(4)}^5} \times {{(4)}^2}] }}{{({2^6} \times {2^4})}} = \dfrac{{[{{({2^2})}^5} \times {{({2^2})}^2}] }}{{({2^6} \times {2^4})}}\]
Using exponent multiplication law \[{a^{mn}} = {({a^m})^n}\] , we get,
 \[ \Rightarrow \dfrac{{[{{(4)}^5} \times {{(4)}^2}] }}{{({2^6} \times {2^4})}} = \dfrac{{[{2^{10}} \times {2^4}] }}{{({2^6} \times {2^4})}}\]
Using exponent multiplication law \[{a^m} \times {a^n} = {a^{m + n}}\] , we get,
 \[ \Rightarrow \dfrac{{[{{(4)}^5} \times {{(4)}^2}] }}{{({2^6} \times {2^4})}} = \dfrac{{{2^{10 + }}^4}}{{{2^{6 + 4}}}}\]
 \[ \Rightarrow \dfrac{{[{{(4)}^5} \times {{(4)}^2}] }}{{({2^6} \times {2^4})}} = \dfrac{{{2^{14}}}}{{{2^{10}}}}\]
Now using the exponent division law \[\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}\] , we get,
 \[ \Rightarrow \dfrac{{[{{(4)}^5} \times {{(4)}^2}] }}{{({2^6} \times {2^4})}} = {2^{14 - 10}}\]
 \[ \Rightarrow \dfrac{{[{{(4)}^5} \times {{(4)}^2}] }}{{({2^6} \times {2^4})}} = {2^4}\]
Hence the exponential form will be \[{2^4}\] .
So, the correct answer is “ \[{2^4}\] ”.

Note: The easy way to solve the problem is first identifying prime factors in the question and then breaking up the other numbers into the same prime factors already present. We can also solve the given question first then writing the exponential form as follows:
 \[ \Rightarrow \dfrac{{[{{(4)}^5} \times {{(4)}^2}] }}{{({2^6} \times {2^4})}} = \dfrac{{1024 \times 16}}{{64 \times 16}}\]
Solving the multiplication and division, we get,
 \[ \Rightarrow \dfrac{{[{{(4)}^5} \times {{(4)}^2}] }}{{({2^6} \times {2^4})}} = 16\]
Now \[16 = 2 \times 2 \times 2 \times 2 = {2^4}\]
Hence the exponential form will be \[{2^4}\] .