To solve the problem, let's consider the given expressions and work through them step-by-step.
We are given:
[tex]\[ 4^r = a \][/tex]
[tex]\[ 2^r = b \][/tex]
We need to find the value of [tex]\( 4^{r+1} \cdot 2^{r+1} \)[/tex].
First, let's express [tex]\( 4^{r+1} \)[/tex] in terms of [tex]\( a \)[/tex]:
[tex]\[ 4^{r+1} = 4^r \cdot 4 = a \cdot 4 \][/tex]
Next, let's express [tex]\( 2^{r+1} \)[/tex] in terms of [tex]\( b \)[/tex]:
[tex]\[ 2^{r+1} = 2^r \cdot 2 = b \cdot 2 \][/tex]
Now, we need to find the product [tex]\( 4^{r+1} \cdot 2^{r+1} \)[/tex]:
[tex]\[ 4^{r+1} \cdot 2^{r+1} = (a \cdot 4) \cdot (b \cdot 2) \][/tex]
Simplify the product:
[tex]\[ (a \cdot 4) \cdot (b \cdot 2) = a \cdot b \cdot 4 \cdot 2 \][/tex]
[tex]\[ = a \cdot b \cdot 8 \][/tex]
[tex]\[ = 8 \cdot a \cdot b \][/tex]
Therefore, the expression [tex]\( 4^{r+1} \cdot 2^{r+1} \)[/tex] simplifies to [tex]\( 8 \cdot a \cdot b \)[/tex].
Thus, the correct answer is:
D) [tex]\( 8 a b \)[/tex]
