Find the missing value for 4−3⋅4?=4134^{-3}\cdot4^?=4^{13}4

−3
⋅4
?
=4
13

1 answer

To solve for the missing exponent in the equation \( 4^{-3} \cdot 4^? = 4^{13} \), we can use the laws of exponents.

According to the law of exponents, when multiplying like bases, you add the exponents:

\[ 4^{-3} \cdot 4^? = 4^{-3 + ?} \]

So, we can rewrite the equation as:

\[ 4^{-3 + ?} = 4^{13} \]

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

\[ -3 + ? = 13 \]

To solve for \( ? \), we can add 3 to both sides:

\[ ? = 13 + 3 \] \[ ? = 16 \]

Thus, the missing value is:

\[ \boxed{16} \]