Question

How do you simplify ${{p}^{7}}.{{q}^{5}}.{{p}^{6}}$?

Answer 1

Hint: We first explain the process of exponents and indices. We find the general form. Then we explain the different binary operations on exponents. Finally, we find the indices number for the multiplication of ${{p}^{7}}.{{q}^{5}}.{{p}^{6}}$ and express it in its simplified form.

Complete step-by-step answer:
We know the exponent form of the number $a$ with the exponent being $n$ can be expressed as ${{a}^{n}}$.
The simplified form of the expression ${{a}^{n}}$ can be written as the multiplied form of number $a$ of n-times.
Therefore, ${{a}^{n}}=\underbrace{a\times a\times a\times ....\times a\times a}_{n-times}$.
The value of $n$ can be any number belonging to the domain of real numbers.
Similarly, the value of $a$ can be any number belonging to the domain of real numbers.
In case the value of $n$ becomes negative, the value of the exponent takes its inverse value.
The formula to express the form is ${{a}^{-n}}=\dfrac{1}{{{a}^{n}}},n\in {{\mathbb{R}}^{+}}$.
The multiplication of these exponents works as the addition of those indices.
For example, we take two exponential expressions where the exponents are $m$ and $n$.
Let the numbers be ${{a}^{m}}$ and ${{a}^{n}}$. We take multiplication of these numbers.
The indices get added. So, ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$.
Now for ${{p}^{7}}.{{q}^{5}}.{{p}^{6}}$, we can simplify the indices for $p$.
So, ${{p}^{7}}\times {{p}^{6}}={{p}^{7+6}}={{p}^{13}}$.
The simplified form is ${{p}^{7}}.{{q}^{5}}.{{p}^{6}}={{p}^{13}}{{q}^{5}}$.

Note: The sum or subtraction of indices in case of multiplication and division is only possible when the base values are equal. We cannot simplify the indices for $p$ and $q$. We have to keep them in their multiplied form.