Let's start by examining the equation:
[tex]\[ 4^3 \cdot 4^{\square} = 4^8 \][/tex]
Notice that we're working with powers of 4. When multiplying terms with the same base, we can add their exponents:
[tex]\[ 4^a \cdot 4^b = 4^{a+b} \][/tex]
Using this property, we rewrite the original equation by adding the exponents on the left-hand side:
[tex]\[ 4^3 \cdot 4^{\square} = 4^{3 + \square} \][/tex]
We need this to equal \( 4^8 \):
[tex]\[ 4^{3 + \square} = 4^8 \][/tex]
Since the bases are the same (both are 4), we can set the exponents equal to each other:
[tex]\[ 3 + \square = 8 \][/tex]
To find the value of \( \square \), we solve for \( \square \):
[tex]\[ \square = 8 - 3 \][/tex]
[tex]\[ \square = 5 \][/tex]
Thus, the completed equation is:
[tex]\[ 4^3 \cdot 4^5 = 4^8 \][/tex]
And the value for \( \square \) is:
[tex]\[ \square = 5 \][/tex]
