Question

What is the value of the expression \[\dfrac{{{2^6}}}{{{2^2}}}\] ?

Answer 1

Hint: Here we have to solve the exponential function. We will use the law of exponents to solve the given problem. When the bases are the same and the operation between the bases is division, then the power of the denominator is subtracted from the power of the numerator and can be expressed as $\dfrac{{{r^n}}}{{{r^m}}} = {r^{n - m}}$ where $r$ is the base and $m$ and $n$ are the exponents.

Complete step by step answer:
Exponential function can be defined as the function which is in the form $f(x) = {a^x}$ where $x$ is the variable and $a$ is constant and also known as the base of the function and it should always be greater than zero. When a number is multiplied with itself $n$ times, the number is raised to the power of $n$. Let the number be $a$ that is multiplied with itself $n$ times it can be written as ${a^n}$ .

In this question, $2$ is raised to the power $6$ in the numerator and $2$ in the denominator. We will solve the given exponential using the laws of exponents which states that if bases are the same and the operation between the bases is division, then the power of the denominator is subtracted from the power of the numerator and can be expressed as $\dfrac{{{r^n}}}{{{r^m}}} = {r^{n - m}}$ where $r$ is the base and $m$ and $n$ are the exponents.

The given exponential function is \[\dfrac{{{2^6}}}{{{2^2}}}\]. Using the above law of exponents in the given function. we get,
\[ \Rightarrow \dfrac{{{2^6}}}{{{2^2}}} = {2^6} - {2^2}\]
Solving the power of the base. We get,
$ \Rightarrow {2^4}$
Expanding the power of $2$. We get,
$ \Rightarrow {2^4} = 2 \times 2 \times 2 \times 2$
\[ \therefore {2^6} = 16\]

Hence, the value of the expression \[\dfrac{{{2^6}}}{{{2^2}}}\] is $16$.

Note: In this question, we are given an exponential function. When a number is raised to some power, the number is said to be in exponential form, for example ${a^n}$ is an exponential function. The most commonly used exponential function is the base of the transcendental number denoted by $e$, which is approximately equal to the value of $2.71828$.