Question

Simplify and write the answer in the exponential form: \[\left[ {{{\left( {{2^2}} \right)}^3} \times {3^6}} \right] \times {5^6}\]

Answer 1

Hint: To solve this question first we assume a variable equal to the given expression. Then we solve the powers of the curved bracket and then we use the property of the exponent that numbers are different and the power is the same. Then we multiply all the numbers and put power as it is from here we got in the exponential form and by multiplying that we got the normal form also.

Complete step-by-step solution:
Let \[x = \left[ {{{\left( {{2^2}} \right)}^3} \times {3^6}} \right] \times {5^6}\]
To solve further we power on 2.
If a power is applied on a number and then again power is used then both the power can get multiplied.
\[{\left( {{a^b}} \right)^c} = {a^{bc}}\] here b and c both are multiplied. Example. \[{\left( {{2^2}} \right)^3} = {2^{2 \times 3}}\]
\[x = \left[ {{2^6} \times {3^6}} \right] \times {5^6}\]
Now we can remove brackets because all are joined by the multiplication sign.
\[x = {2^6} \times {3^6} \times {5^6}\]
Using the property of exponent.
\[{a^b} \times {c^b} = {\left( {ac} \right)^b}\]
Here in particular, there are three numbers so we multiply all the three numbers.
\[x = {\left( {2 \times 3 \times 5} \right)^6}\]
On further solving we get
\[x = {\left( {30} \right)^6}\]
The value of the given expression in the exponential form.
\[\left[ {{{\left( {{2^2}} \right)}^3} \times {3^6}} \right] \times {5^6} = {\left( {30} \right)^6}\]
The value of the given expression in the normal form.
\[\left[ {{{\left( {{2^2}} \right)}^3} \times {3^6}} \right] \times {5^6} = 729000000\]

Note: To solve this question we use the property of exponent. The properties are the multiplication of different numbers with the same power then both the numbers are multiplied. Example- \[{a^b} \times {c^b} = {\left( {ac} \right)^b}\]. The next property is different power on the same number and the got number is multiplied by each other then the powers are added in the final expression. Example \[{a^b} \times {a^c} = {a^{\left( {b + c} \right)}}\]. Like these, there are many properties which are of exponent which you have to remember. In this type of question, students are directly calculating the powers and multiplying them. From there they got simple values but if asked to represent in the exponential form then they are unable to change in exponential form.