Question

How do you simplify \[{3^{ - 2}}\]?

Answer 1

Hint: Here, the power of the number 3 is negative, so we will take reciprocal of the given number using the negative exponential rule and write the base 3 with a positive power in the denominator. Then, we will simplify the expression to get the required answer.

Formula Used:
\[{a^{ - b}} = \dfrac{1}{{{a^b}}}\]

Complete step-by-step answer:
According to the negative exponent rule, if we have a negative exponent in the numerator then, it gets moved to the denominator and the exponent becomes positive.
Thus, in order to simplify \[{3^{ - 2}}\], we will use the identity \[{a^{ - b}} = \dfrac{1}{{{a^b}}}\].
Thus, we get,
\[{3^{ - 2}} = \dfrac{1}{{{3^2}}}\]
Now, we know that \[{3^2} = 3 \times 3 = 9\]
Therefore, we get,
\[{3^{ - 2}} = \dfrac{1}{9}\]

Hence, the simplified form of \[{3^{ - 2}}\] is \[\dfrac{1}{9}\]
Thus, this is the required answer.

Additional information:
When we are required to show an exponent that is present in the denominator of a fraction, then, instead of writing the fraction, we use negative exponents and hence, reverse the denominator and write \[\dfrac{1}{{{a^m}}} = {a^{ - m}}\]. This is also known as the negative exponent rule. We can easily represent a large number using exponents. When a number has an exponent as 0, then its value is equal to 1.

Note:
An expression that represents the repeated multiplication of the same number is known as power. Whereas, when a number is written with power then the power becomes the exponent of that particular number. It shows the number of times that particular number will be multiplied by itself. Hence, whenever we are given the multiplication of the same numbers then, we can express that number with an exponent.