Question

Simplify and express as positive indices.
${\left( {{x^{\left( n \right)}}{y^{ - m}}} \right)^4} \times {\left( {{x^3}{y^{ - 2}}} \right)^{ - n}}$

Answer 1

Hint: Index of a variable (or a constant) is a value that is raised to the power of the variable. The indices are also known as powers or exponents. It shows the number of times a given number has to be multiplied. It is represented in the form
${a^{\left( n \right)}} = a \times a \times ..... \times a$ (n times a).
Here, a is the base & n is the index. It is a compressed method of writing big numbers and calculations.
To simplify the given example we have to use the multiplication rule of indices.
Multiplication rule of indices is,
${x^{\left( m \right)}}{x^{\left( n \right)}} = {x^{m + n}}$

Complete step-by-step solution:
We have given,
${\left( {{x^{\left( n \right)}}{y^{ - m}}} \right)^4} \times {\left( {{x^3}{y^{ - 2}}} \right)^{ - n}}$
When a variable with some index is again raised with a different index, and then both the indices are multiplied together, raise to the power of the same base.
It means ${\left( {{x^{\left( m \right)}}} \right)^n} = {x^{m \times n}}$
 $ \Rightarrow \left( {{x^{4n}}{y^{ - 4m}}} \right) \times \left( {{x^{ - 3n}}{y^{ - 2\left( { - n} \right)}}} \right)$
Simplify it.
$ \Rightarrow \left( {{x^{4n}}{y^{ - 4m}}} \right) \times \left( {{x^{ - 3n}}{y^{2n}}} \right)$
Now, we have to multiply the same bases.
$ \Rightarrow \left( {{x^{4n}} \times {x^{ - 3n}}} \right)\left( {{y^{ - 4m}} \times {y^{2n}}} \right)$
Here we will use the multiplication rule which is explained in the hint.
So, we will get,
$ \Rightarrow \left( {{x^{4n}}^{ + \left( { - 3n} \right)}} \right)\left( {{y^{ - 4m}} \times {y^{2n}}} \right)$
Let us simplify the index part of base x.
$ \Rightarrow \left( {{x^{\left( n \right)}}} \right)\left( {{y^{ - 4m}} \times {y^{2n}}} \right)$
For negative indices, we can use the reciprocal rule. If the index is the negative value, then it can be shown as the reciprocal of the positive index raised to the same variable.
It means,
${a^{\left( { - m} \right)}} = \dfrac{1}{{{a^{\left( m \right)}}}}$
So, we get,
$ \Rightarrow \left( {{x^{\left( n \right)}}} \right)\left( {\dfrac{1}{{{y^{4m}}}} \times {y^{2n}}} \right)$
Simplify it.
$ \Rightarrow \left( {{x^{\left( n \right)}}} \right)\left( {\dfrac{{{y^{2n}}}}{{{y^{4m}}}}} \right)$
Let us simplify it again.
$ \Rightarrow \left( {\dfrac{{{x^{\left( n \right)}}{y^{2n}}}}{{{y^{4m}}}}} \right)$
This is the final answer with positive indices.

Note: If simplifying this kind of problem, the student should be familiar with all the laws of indices.
Applications of indices:
1) Exponents are commonly used in computer games.
2) Power of 2 exponents is the base of all computing which is done in what is called, binary (or base of two numbers).