Question

How do you simplify $4{{n}^{4}}2{{n}^{-3}}$ and write it using only positive exponents.

Answer 1

Hint: Now to simplify the expression we will first multiply the coefficients or the constants in the expression and simplify. Now we know that ${{a}^{m}}{{a}^{n}}={{a}^{m+n}}$ hence using this property of indices we will simplify the obtained expression.

Complete step by step solution:
Now let us first understand the concept of indices.
Now an index is a value which is raised to a number or a variable.
The number which is raised is also called power of number or variable.
Now the index or power is nothing but a number which denotes the number of times a variable or term is multiplied by itself.
For example let us consider ${{a}^{3}}$ . Now here we have the power as 3. Hence a is multiplied to itself 3 times. Hence we can write ${{a}^{3}}=a\times a\times a$ .
Now let us understand what happens when we have negative power in indices.
If we have a negative number in the index then the term can be shifted into a denominator and the index is written as a positive number of the same value.
Now in indices if we have ${{a}^{m}}.{{a}^{n}}$ then the value is equal to ${{a}^{m+n}}$
We will use this rule to solve the following problem.
Now consider the given example $4{{n}^{4}}2{{n}^{-3}}$
Now simplifying the above expression we get, $8{{n}^{4+\left( -3 \right)}}=8{{n}^{\left( 4-3 \right)}}=8{{n}^{1}}$
Hence the given expression can be written as $8n$ .

Note:
Now note that the index of a number can be negative, positive or 0. If we have a negative number in the index then we can convert it as ${{a}^{-n}}=\dfrac{1}{{{a}^{n}}}$ . Now note that for any number a ${{a}^{0}}=1$ .
Now of the index is a fraction then we have ${{a}^{\dfrac{p}{q}}}=\sqrt[q]{{{a}^{p}}}$ .