Complete the equation.

[tex]\[
\begin{array}{l}
4^3 \cdot 4^{\square}=4^8 \\
\square = \square
\end{array}
\][/tex]

Solve for the missing exponent.

[tex]\[
4^3 \cdot 4^{\square}=4^8
\][/tex]

[tex]\[
\square = 5
\][/tex]



Answer :

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]