Let's solve the problem step by step. We need to find the value of [tex]\(\sqrt[3]{2 a^2 b^4} + \frac{a-c}{(b+c)^2}\)[/tex] given that [tex]\(a = 4\)[/tex], [tex]\(b = 2\)[/tex], and [tex]\(c = -5\)[/tex].
### Step 1: Simplify the first term [tex]\(\sqrt[3]{2 a^2 b^4}\)[/tex]
Substitute the given values for [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
\sqrt[3]{2 \cdot (4)^2 \cdot (2)^4} = \sqrt[3]{2 \cdot 16 \cdot 16} = \sqrt[3]{2 \cdot 256}
\][/tex]
First, multiply [tex]\(2\)[/tex] by [tex]\(256\)[/tex]:
[tex]\[
2 \cdot 256 = 512
\][/tex]
Next, take the cube root of [tex]\(512\)[/tex]:
[tex]\[
\sqrt[3]{512} = 8
\][/tex]
So, the value of the first term is:
[tex]\[
\sqrt[3]{2 a^2 b^4} = 8
\][/tex]
### Step 2: Simplify the second term [tex]\(\frac{a-c}{(b+c)^2}\)[/tex]
Substitute the given values for [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\frac{4 - (-5)}{(2 + (-5))^2} = \frac{4 + 5}{(2 - 5)^2} = \frac{9}{(-3)^2} = \frac{9}{9}
\][/tex]
Simplify the expression:
[tex]\[
\frac{9}{9} = 1
\][/tex]
So, the value of the second term is:
[tex]\[
\frac{a-c}{(b+c)^2} = 1
\][/tex]
### Step 3: Add the simplified terms together
Add the two results obtained from steps 1 and 2:
[tex]\[
8 + 1 = 9
\][/tex]
Thus, the value of the expression [tex]\(\sqrt[3]{2 a^2 b^4} + \frac{a-c}{(b+c)^2}\)[/tex] is:
[tex]\[
9
\][/tex]
