Question

In \[{8^9},9\] is
(A) \[{\text{Exponent}}\]
(B) \[{\text{Base }}\]
(C) \[{\text{Exponent and power}}\]
(D) \[{\text{None of these}}\]

Answer 1

Hint: We are using the concept of powers, bases and some identities, to solve this problem. Also, we will know about some formulas while solving this problem. We will also know a little about surds and their identities too.

Complete step by step answer:
 Generally, an exponent refers to the number of times a number is multiplied to itself. For example, \[{{\text{5}}^3}\] means that 5 is multiplied by itself three times i.e. \[5 \times 5 \times 5\] . So, in a term of type \[{a^b}\] , \[a\] is called the base and \[b\] is called the exponent or power. Base can be a rational number or an irrational number or a complex number too. For example, \[{i^2} = - 1\] .
In most general cases, the base will be \[e\] where, \[e = 2.17\] .
Most generally, we use the term “power” in most of the cases.
Let us know about some identities of exponents.
The first identity is \[{a^m} \times {a^n} = {a^{(m + n)}}\]
And similarly, \[\dfrac{{{a^m}}}{{{a^n}}} = {a^{(m - n)}}\]
And \[{\left( {{a^m}} \right)^n} = {a^{mn}}\]
If an exponent or a power is a fraction, then the term is called a surd. For example, \[{a^{\dfrac{1}{b}}}\] is a surd.
Remember that, a surd can also be written like this \[{a^{\dfrac{1}{b}}} = \sqrt[b]{a}\]
We can solve many sums and problems using these identities.
From all the information, we can conclude that, in \[{8^9}\] , \[9\] is called the Exponent and also a Power.

So, the correct answer is “Option C”.

Note:
Note that any number to the power zero is equal to one i.e. \[{a^0} = 1\] . Using these exponents and surds, we can also evaluate logarithms also. A power can also be a negative number and the identity relating to it will be as follows….\[{a^{ - m}} = \dfrac{1}{{{a^m}}}\] . But make sure that a power shouldn’t be infinity which is not defined. And also remember that, an exponent is also called power.