Question

How do you evaluate \[{4^2} \cdot {4^2}\] ?

Answer 1

Hint: Given expression is \[{4^2} \cdot {4^2}\] . We have to evaluate the expression and find its value. To approach the solution, we have to use mathematical identities instead of directly calculating the expression in the first place. Here, we can use the mathematical identities \[{a^n} \cdot {b^n} = {\left( {ab} \right)^n}\] and \[{\left( {{a^n}} \right)^m} = {a^{mn}}\] to further evaluate the given expression.

Complete step by step solution:
Given expression is
 \[{4^2} \cdot {4^2}\]
Since \[4 = 2 \times 2 = {2^2}\]
We can write the above expression as
 \[ \Rightarrow {\left( {{2^2}} \right)^2} \cdot {\left( {{2^2}} \right)^2}\]
Now, using the mathematical identity \[{\left( {{a^n}} \right)^m} = {a^{mn}}\]
We can write the above expression as
 \[ \Rightarrow {\left( 2 \right)^{2 \cdot 2}} \cdot {\left( 2 \right)^{2 \cdot 2}}\]
\[ \Rightarrow {\left( 2 \right)^4} \cdot {\left( 2 \right)^4}\]
Again, using the mathematical identity \[{a^n} \cdot {b^n} = {\left( {ab} \right)^n}\]
We can write the above expression as
\[ \Rightarrow {\left( {2 \cdot 2} \right)^4}\]
Since \[2 \cdot 2 = {2^2}\]
So we can write
\[ \Rightarrow {\left( {{2^2}} \right)^4}\]
Again using the mathematical identity \[{\left( {{a^n}} \right)^m} = {a^{mn}}\]
We can write the above expression as
\[ \Rightarrow {\left( 2 \right)^{2 \cdot 4}}\]
\[ \Rightarrow {\left( 2 \right)^8}\]
Now \[{\left( 2 \right)^8}\] can be written as
\[ \Rightarrow 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\]
\[ \Rightarrow 8 \times 8 \times 2 \times 2\]
So we can write it as,
\[ \Rightarrow 64 \times 2 \times 2\]
So we get,
\[ \Rightarrow 128 \times 2\]
Again, that gives
\[ \Rightarrow 256\]
That is the required evaluation of the given expression.
Therefore, the evaluated value of \[{4^2} \cdot {4^2}\] is \[256\].

Note:
In an expression such as \[{a^b}\] , \[a\] is called base whereas \[b\] is called the exponent. . For example, in the case of \[{4^2}\], the number \[4\] is called the base, and the number \[2\] is the exponent. Any two or more such expressions can be multiplied or divided by using their mathematical identities for exponents and power.
An expression which represents the repeated multiplication of the same factor is called a power. An Exponent corresponds to the number of times the base is utilized as a factor in an expression.
So exponent is also a method to express a large set of natural numbers which cannot be represented in general form. If we have a large number supposed to be \[100000000\], we can exponentially represent the number as \[{10^8}\].
So the exponential notation is also a way to shorten the large numbers so that the calculation can be convenient and easy to express.