Question

How do you write the expression \[4.4.4.4.4.4\] using exponents?

Answer 1

Hint:We need to write \[4.4.4.4.4.4\] in exponential form. We should first see the general form of writing the expressions of this type in exponential form and then we will try to use that on the given expression. Let us consider \[\underbrace {a.a.a.a.a.a.a.a.......a}_n\], which means that the number \[a\] is multiplied \[n\] times. Now, we write this as \[{a^n}\] in exponential form.

Complete step by step answer:
We are given \[4.4.4.4.4.4\] and we need to write this expression in exponential form.
Let us first see how many \[4\] are there in \[4.4.4.4.4.4\].
We see that \[4\] is multiplied \[6\] times.
Hence, we can write
\[ \Rightarrow 4.4.4.4.4.4 = \underbrace {4.4.4.4.4.4}_6\]
Now, using \[\underbrace {a.a.a.a.a.a.a.a.......a}_n = {a^n}\], we have
\[ \Rightarrow 4.4.4.4.4.4 = \underbrace {4.4.4.4.4.4}_6 = {4^6}\]
Now, we know that \[4 = {2^2}\]. So, we get
\[ \Rightarrow 4.4.4.4.4.4 = \underbrace {4.4.4.4.4.4}_6 = {4^6} = {\left( {{2^2}} \right)^6}\]
Now, using the property \[{\left( {{a^b}} \right)^c} = {a^{bc}}\], we have
\[ \Rightarrow 4.4.4.4.4.4 = \underbrace {4.4.4.4.4.4}_6 = {4^6} = {\left( {{2^2}} \right)^6}\]
\[ \Rightarrow 4.4.4.4.4.4 = {4^6} = {\left( {{2^2}} \right)^6} = {2^{2 \times 6}}\]
\[ \Rightarrow 4.4.4.4.4.4 = {4^6} = {\left( {{2^2}} \right)^6} = {2^{12}}\]
\[ \therefore 4.4.4.4.4.4 = {4^6} = {2^{12}}\]
Hence, we can write \[4.4.4.4.4.4\] as \[{4^6}\] or \[{2^{12}}\] in the exponential form.

Note: The most common mistake we make while solving this type of problem is not counting the number of times the given number is multiplied correctly. Once we count the wrong number, we will reach the wrong answer. Also, to solve such types of questions, we need to be very thorough with all the properties of exponents.