Question

Evaluate \[13\] to the power of \[ - 2\] ?

Answer 1

Hint: In this problem we need to evaluate the value of \[13\] raised to the power of \[ - 2\] . \[13\] to the negative \[ - 2\] power is conventionally written as \[{13^{ - 2}}\] . Now, we know that \[{a^{ - n}} = \dfrac{1}{{{a^n}}}\] .Hence using this property we will first write \[{13^{ - 2}}\] with positive power. Now we also know that for a positive integer \[n\] we have \[{a^n} = a \times a \times a \times ...n{\text{ }}times\] . So, we will use this property to evaluate the obtained expression and hence find the value of \[{13^{ - 2}}\].

Complete step by step answer:
Consider the given number i.e., \[{13^{ - 2}}\]. Now, since we have \[ - 2\] power, which is negative power, we will first make this power positive. As, we know that
\[{a^{ - n}} = \dfrac{1}{{{a^n}}}\]
So, here \[a = 13\] and \[n = 2\]
Therefore, we get
\[{13^{ - 2}} = \dfrac{1}{{{{13}^2}}}\]
Hence, we make this power positive by taking this to the denominator.

Now, we have \[{13^2}\] in the denominator. Now, we also know that for a positive integer \[n\] we have
\[{a^n} = a \times a \times a \times ...n{\text{ }}times\]
So here, \[a = 13\] and \[n = 2\]
Therefore, we get as
\[{13^2} = 13 \times 13\]
Now, we know that \[13 \times 13 = 169\]
Hence, we have
\[\therefore {13^{ - 2}} = \dfrac{1}{{{{13}^2}}} = \dfrac{1}{{169}}\]

Hence, \[13\] to the power of \[ - 2\] equals \[\dfrac{1}{{169}}\].

Note: Here the number \[13\] is called the base, and the number \[ - 2\] is called the exponent or index. This question can also be understood in a simple way like: As the exponent is a negative integer, so exponentiation means the reciprocal of a repeated multiplication. The absolute value of the exponent of the number \[ - n\] i.e., \[n\] denotes how many times we have to multiply the base \[a\] ,and the power’s minus sign stands for reciprocal.Thus, we have \[13\] to the power of \[ - 2\] which means the absolute value of the exponent of the number \[ - 2\] i.e., \[2\] denotes how many times we have to multiply the base \[13\] ,and the power’s minus sign stands for reciprocal.Therefore, we get
\[{13^{ - 2}} = \dfrac{1}{{169}}\]