Question

How do you simplify $\dfrac{{3{n^4}}}{{3{n^3}}}$ and write it using only positive exponents?

Answer 1

In this question, we are given a fraction containing numerical values and some unknown variable quantity “n”. For simplifying a fraction, we write the numerator and the denominator as a product of its prime factors, prime factors are the numbers that are only divisible by 1 and themselves.
After doing the prime factorization of the numerator and the denominator, we look for the factors that are present in both the numerator and the denominator, these factors are called the common factors, then we cancel out the common factors until there are no common factors present between the numerator and the denominator.

Complete step by step answer:
$3{n^4}$ can be written as $3 \times n \times n \times n \times n$
And $3{n^3}$ can be written as $3 \times n \times n \times n$
Thus we see that the numerator and the denominator have $3 \times n \times n \times n$ or
$3{n^3}$ as common, so it is canceled out and we get –
$\dfrac{{3{n^4}}}{{3{n^3}}} = n$
The power of n is equal to 1 and it is a positive number.
Hence the simplified form of $\dfrac{{3{n^4}}}{{3{n^3}}}$ is $n$ .

Note: The given question can also be solved using the law which states that when two numbers having the same base but different powers are divided then keeping the base same, we subtract the power of the denominator from the power of the numerator, that is, $\dfrac{{{x^a}}}{{{x^b}}} = {x^{a - b}}$ .
We also note that the exponent of the result obtained is positive, but if the exponent obtained is negative then we write the number as the reciprocal of the given exponent to convert it into positive exponent, that is, ${a^{ - x}} = \dfrac{1}{{{a^x}}}$ .