To solve the problem, we are looking for a number [tex]\( x \)[/tex] such that:
[tex]\[ x \times 13^{-6} = \left(\frac{1}{13}\right)^9 \][/tex]
First, let's rewrite the expression [tex]\(\left(\frac{1}{13}\right)^9\)[/tex] using properties of exponents:
[tex]\[
\left(\frac{1}{13}\right)^9 = 13^{-9}
\][/tex]
So our equation now becomes:
[tex]\[ x \times 13^{-6} = 13^{-9} \][/tex]
To isolate [tex]\( x \)[/tex], we divide both sides of the equation by [tex]\( 13^{-6} \)[/tex]:
[tex]\[
x = \frac{13^{-9}}{13^{-6}}
\][/tex]
Using the properties of exponents ([tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]), we can simplify the right side of the equation:
[tex]\[
x = 13^{-9 + 6}
\][/tex]
[tex]\[
x = 13^{-3}
\][/tex]
Therefore, the number we need to multiply [tex]\( 13^{-6} \)[/tex] by to get [tex]\( \left(\frac{1}{13}\right)^9 \)[/tex] is [tex]\( 13^{-3} \)[/tex].
To provide the numerical value of [tex]\( 13^{-3} \)[/tex]:
[tex]\[ 13^{-3} = \frac{1}{13^3} = \frac{1}{2197} \][/tex]
Converting this division result to a decimal:
[tex]\[
\frac{1}{2197} \approx 0.0004551661356395084
\][/tex]
So, the number by which we need to multiply [tex]\( 13^{-6} \)[/tex] is approximately:
[tex]\[
\boxed{0.0004551661356395084}
\][/tex]
