Question

What is \[a\left( {{a^{n - 1}}} \right)\]?

Answer 1

Hint: Here we will use the concept of exponent to solve the question. First, we will break the term inside the bracket using the rule of exponent. Then we will multiply it by the term outside the bracket and use the property of the product of a power. We will then simplify further to get the desired answer.

Complete step-by-step answer:
As the equation given to us is \[a\left( {{a^{n - 1}}} \right)\] we have to simplify it by using exponent property.
First, we will break the term inside the exponent using the exponent rule \[{x^{y + z}} = {x^y} \cdot {x^z}\].
\[a\left( {{a^{n - 1}}} \right) = a\left( {{a^n} \times {a^{ - 1}}} \right)\]
Multiplying the terms, we get
\[ \Rightarrow a\left( {{a^{n - 1}}} \right) = a \times {a^n} \times {a^{ - 1}}\]
Now using the exponent rule \[{x^{ - 1}} = \dfrac{1}{x}\] in the above equation, we get
\[ \Rightarrow a\left( {{a^{n - 1}}} \right) = a \times a \times \dfrac{1}{a}\]
Multiplying the terms and cancelling the like terms, we get
\[ \Rightarrow a\left( {{a^{n - 1}}} \right) = a\]
Therefore, we get our answer as \[a\].

Note: To solve this question we need to keep in mind the properties of exponents. Some properties of exponent are listed below:
1.When we multiply exponent which same base their power is added also known as product of power
2.When we have a power to the power of an exponent we multiply the power or the exponent also known as the power to power.
3.When we divide the exponent having the same base their power is subtracted also known as the quotient of powers.
4.When there is quotient to a power we give each base its own exponent also known as power of a quotient.
5.When there is a product of a power we give each base its own exponent also known as power of a product.