To find the numerical equivalent of \( 510 \cdot 5^{-6} \), we start by simplifying the expression. The term \( 5^{-6} \) can be rewritten as:
\[ 5^{-6} = \frac{1}{5^6} \]
Substituting this expression back into the original equation gives us:
\[ 510 \cdot 5^{-6} = 510 \cdot \frac{1}{5^6} \]
This can be further simplified to:
\[ \frac{510}{5^6} \]
Now, we need to calculate \( 5^6 \). We can do this step-by-step:
\[ 5^2 = 25 \] \[ 5^3 = 5 \cdot 25 = 125 \] \[ 5^4 = 5 \cdot 125 = 625 \] \[ 5^5 = 5 \cdot 625 = 3125 \] \[ 5^6 = 5 \cdot 3125 = 15625 \]
Now that we know \( 5^6 = 15625 \), we can substitute this value back into our expression:
\[ \frac{510}{5^6} = \frac{510}{15625} \]
Next, we simplify \( \frac{510}{15625} \). We find the greatest common divisor (GCD) of 510 and 15625 to reduce the fraction.
First, we can factor 510:
\[ 510 = 2 \cdot 3 \cdot 5 \cdot 17 \]
Now, let’s check if any of these prime factors are also factors of 15625. Since \( 15625 = 5^6 \), the only common factor is 5. Thus, we can divide both the numerator and the denominator by 5:
\[ \frac{510 \div 5}{15625 \div 5} = \frac{102}{3125} \]
Now, let’s check for any further simplification. The prime factorization of 102 is:
\[ 102 = 2 \cdot 3 \cdot 17 \]
The prime factorization of 3125 is:
\[ 3125 = 5^5 \]
Since there are no common factors left, we have:
\[ \frac{102}{3125} \]
Thus, the numerical equivalent of \( 510 \cdot 5^{-6} \) expressed without any exponent expression or decimals is:
\[ \boxed{\frac{102}{3125}} \]