Let's evaluate the expression step-by-step given the values [tex]\( a = 8 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = 16 \)[/tex].
We are given the expression:
[tex]\[
\frac{2b + 3c^2}{4a^2 - 2b}
\][/tex]
Step 1: Calculate the numerator
To find the numerator, we start by evaluating each part separately and then combine them:
1. Calculate [tex]\( 2b \)[/tex]:
[tex]\[
2b = 2 \times 4 = 8
\][/tex]
2. Calculate [tex]\( 3c^2 \)[/tex]:
[tex]\[
c^2 = 16^2 = 256
\][/tex]
[tex]\[
3c^2 = 3 \times 256 = 768
\][/tex]
3. Sum the results to get the numerator:
[tex]\[
2b + 3c^2 = 8 + 768 = 776
\][/tex]
Thus, the numerator is 776.
Step 2: Calculate the denominator
To find the denominator, follow the same approach by evaluating each part separately and then combine them:
1. Calculate [tex]\( 4a^2 \)[/tex]:
[tex]\[
a^2 = 8^2 = 64
\][/tex]
[tex]\[
4a^2 = 4 \times 64 = 256
\][/tex]
2. Calculate [tex]\( -2b \)[/tex]:
[tex]\[
-2b = -2 \times 4 = -8
\][/tex]
3. Subtract the results to get the denominator:
[tex]\[
4a^2 - 2b = 256 - 8 = 248
\][/tex]
Thus, the denominator is 248.
Step 3: Evaluate the expression
Finally, divide the numerator by the denominator to evaluate the expression:
[tex]\[
\frac{2b + 3c^2}{4a^2 - 2b} = \frac{776}{248} = 3.129032258064516
\][/tex]
So, the evaluated expression value is approximately [tex]\( 3.129 \)[/tex].
