To evaluate the expression [tex]\(\sqrt[3]{2 a^2 b^4} + \frac{a - c}{(b + c)^2}\)[/tex] given the values [tex]\(a = 4\)[/tex], [tex]\(b = 2\)[/tex], and [tex]\(c = -5\)[/tex], we can break it down into two parts and calculate each term separately.
### Step 1: Compute [tex]\(\sqrt[3]{2 a^2 b^4}\)[/tex]
First, we need to substitute the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the expression [tex]\( \sqrt[3]{2 a^2 b^4} \)[/tex]:
[tex]\[
\sqrt[3]{2 \times 4^2 \times 2^4}
\][/tex]
Calculate the powers and products inside the cubic root:
[tex]\[
4^2 = 16 \quad \text{and} \quad 2^4 = 16
\][/tex]
[tex]\[
2 \times 16 \times 16 = 2 \times 256 = 512
\][/tex]
Now calculate the cubic root of 512:
[tex]\[
\sqrt[3]{512} = 8
\][/tex]
So, we have:
[tex]\[
\sqrt[3]{2 a^2 b^4} = 8
\][/tex]
### Step 2: Compute [tex]\(\frac{a - c}{(b + c)^2}\)[/tex]
Next, we substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the expression [tex]\(\frac{a - c}{(b + c)^2}\)[/tex]:
[tex]\[
\frac{4 - (-5)}{(2 + (-5))^2}
\][/tex]
Simplify the numerator and the denominator separately:
[tex]\[
4 - (-5) = 4 + 5 = 9
\][/tex]
[tex]\[
2 + (-5) = 2 - 5 = -3
\][/tex]
[tex]\[
(-3)^2 = 9
\][/tex]
So, the expression becomes:
[tex]\[
\frac{9}{9} = 1
\][/tex]
### Step 3: Sum the results of the two terms
Now, we add the results from Step 1 and Step 2:
[tex]\[
8 + 1 = 9
\][/tex]
Thus, the value of [tex]\(\sqrt[3]{2 a^2 b^4} + \frac{a - c}{(b + c)^2}\)[/tex] when [tex]\(a = 4\)[/tex], [tex]\(b = 2\)[/tex], and [tex]\(c = -5\)[/tex] is:
[tex]\[
\boxed{9}
\][/tex]
