Question

How do you simplify $\dfrac{{{4}^{6}}}{{{4}^{8}}}$ ?

Answer 1

Hint: To simplify the above expression $\dfrac{{{4}^{6}}}{{{4}^{8}}}$, we are going to use the property which says that if the two same numbers are written in division form and both the numbers in the numerator and denominator have some power, let us say m and n then we can write the division as follows:
$\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$ . And hence, simplify the above division.

Complete step-by-step answer:
The division expression given in the above problem is as follows:
$\dfrac{{{4}^{6}}}{{{4}^{8}}}$
Now, the above division is of the following form:
$\dfrac{{{a}^{m}}}{{{a}^{n}}}$
And we know that when in the numerator and the denominator base is same and exponents are different then the above division is reduced to:
$\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$
Applying the above division formula in the given expression then substituting $a=4$, $m=6\And n=8$ we get,
$\Rightarrow \dfrac{{{4}^{6}}}{{{4}^{8}}}={{4}^{6-8}}$
Subtracting the exponents in 4 we get,
$\Rightarrow {{4}^{-2}}$
Rearranging the power of the above number we get,
$\Rightarrow {{\left( {{4}^{2}} \right)}^{-1}}$
We know that the second power of 4 is equal to:
${{4}^{2}}=4\times 4=16$
Substituting the above second power of 2 in ${{\left( {{4}^{2}} \right)}^{-1}}$ we get,
${{\left( 16 \right)}^{-1}}$
Now, we know that when the exponent negative of 1 then it represents the reciprocal of the number so we can write the above expression as the reciprocal of 16.
$\dfrac{1}{16}$
Hence, the simplification of the above expression is equal to $\dfrac{1}{16}$.

Note: The mistake that could happen in the above problem is that while subtracting the exponents which are the following step in the above:
$\Rightarrow \dfrac{{{4}^{6}}}{{{4}^{8}}}={{4}^{6-8}}$
Now, in this step, you might have the tendency to subtract a smaller number from the larger number. This is the general tendency we have so on subtracting smaller number from the larger number you will get,
${{4}^{8-6}}={{4}^{2}}$
And simplification of the above expression will give you:
16
And the above answer is the wrong answer so make sure you won’t make such mistakes in the examination.