Find the missing value for 5−5⋅5?=5125^{-5}\cdot5^?=5^{12}5^−5⋅5^?=5^12

Answer 1

To solve the equation \( 5^{-5} \cdot 5^{?} = 5^{12} \), we can use the laws of exponents. Specifically, we know that when multiplying powers with the same base, we can add the exponents:

\[ 5^{-5} \cdot 5^{?} = 5^{-5 + ?} \]

Setting this equal to \( 5^{12} \) gives us:

\[ 5^{-5 + ?} = 5^{12} \]

Since the bases are the same, we can set the exponents equal to each other:

\[ -5 + ? = 12 \]

Now, we can solve for \( ? \):

\[ ? = 12 + 5 \]

\[ ? = 17 \]

Thus, the missing value is \( \boxed{17} \).