To simplify the expression [tex]\( 12^{-2} \)[/tex]:
1. Substitute the base and the exponent into the expression: [tex]\[ 12^{-2} \][/tex]
2. Recall that a negative exponent means taking the reciprocal of the base and then raising it to the corresponding positive exponent. Specifically, for any nonzero number [tex]\( a \)[/tex] and integer [tex]\( n \)[/tex], [tex]\( a^{-n} = \frac{1}{a^n} \)[/tex].
3. Apply the property of exponents to our expression: [tex]\[ 12^{-2} = \frac{1}{12^2} \][/tex]
4. Calculate [tex]\( 12^2 \)[/tex] : [tex]\[ 12^2 = 144 \][/tex]
5. Substitute back into the expression: [tex]\[ \frac{1}{12^2} = \frac{1}{144} \][/tex]
So, the simplest form of the expression [tex]\( 12^{-2} \)[/tex] is [tex]\( \frac{1}{144} \)[/tex].
The decimal form of [tex]\( \frac{1}{144} \)[/tex] is approximately: [tex]\[ 0.006944444444444444 \][/tex]
Therefore, [tex]\( 12^{-2} \)[/tex] in simplest form is [tex]\( \frac{1}{144} \)[/tex], and the decimal representation of this fraction is [tex]\( 0.006944444444444444 \)[/tex].
